Artificial Transmission Lines for RF and Microwave Applications 1118487605, 9781118487600

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ARTIFICIAL TRANSMISSION LINES FOR RF AND MICROWAVE APPLICATIONS

WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANG, Series Editor Texas A&M University

A complete list of the titles in this series appears at the end of this volume.

ARTIFICIAL TRANSMISSION LINES FOR RF AND MICROWAVE APPLICATIONS

FERRAN MARTÍN

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Martín, Ferran, 1965– Artificial transmission lines for RF and microwave applications / Ferran Martín. pages cm. Includes bibliographical references and index. ISBN 978-1-118-48760-0 (hardback) 1. Radio lines. 2. Microwave transmission lines. I. Title. TK6565.T73M37 2015 621.3841 3–dc23 2015007897 Set in 10/12pt Times by SPi Global, Pondicherry, India Printed in the United States of America 10

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1 2015

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To Anna, Alba and Arnau

CONTENTS

Preface

xiii

Acknowledgments

xvii

1 Fundamentals of Planar Transmission Lines

1

1.1 Planar Transmission Lines, Distributed Circuits, and Artificial Transmission Lines, 1 1.2 Distributed Circuit Analysis and Main Transmission Line Parameters, 5 1.3 Loaded (Terminated) Transmission Lines, 8 1.4 Lossy Transmission Lines, 16 1.4.1 Dielectric Losses: The Loss Tangent, 19 1.4.2 Conductor Losses: The Skin Depth, 25 1.5 Comparative Analysis of Planar Transmission Lines, 28 1.6 Some Illustrative Applications of Planar Transmission Lines, 31 1.6.1 Semilumped Transmission Lines and Stubs and Their Application to Low-Pass and Notch Filters, 31 1.6.2 Low-Pass Filters Based on Richard’s Transformations, 39 1.6.3 Power Splitters Based on λ/4 Lines, 40 1.6.4 Capacitively Coupled λ/2 Resonator Bandpass Filters, 42 References, 44 2 Artificial Transmission Lines based on Periodic Structures 2.1 Introduction and Scope, 47 2.2 Floquet Analysis of Periodic Structures, 48

47

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2.3 The Transfer Matrix Method, 53 2.3.1 Dispersion Relation, 54 2.3.2 Bloch Impedance, 56 2.3.3 Effects of Asymmetry in the Unit Cell through an Illustrative Example, 60 2.3.4 Comparison between Periodic Transmission Lines and Conventional Lines, 62 2.3.5 The Concept of Iterative Impedance, 63 2.4 Coupled Mode Theory, 64 2.4.1 The Cross-Section Method and the Coupled Mode Equations, 65 2.4.2 Relation between the Complex Mode Amplitudes and S-Parameters, 69 2.4.3 Approximate Analytical Solutions of the Coupled Mode Equations, 71 2.4.4 Analytical Expressions for Relevant Parameters of EBG Periodic Structures, 77 2.4.5 Relation between the Coupling Coefficient and the S-Parameters, 79 2.4.6 Using the Approximate Solutions of the Coupled Mode Equations, 80 2.5 Applications, 86 2.5.1 Applications of Periodic Nonuniform Transmission Lines, 86 2.5.1.1 Reflectors, 86 2.5.1.2 High-Q Resonators, 92 2.5.1.3 Spurious Suppression in Planar Filters, 93 2.5.1.4 Harmonic Suppression in Active Circuits, 95 2.5.1.5 Chirped Delay Lines, 99 2.5.2 Applications of Reactively Loaded Lines: The Slow Wave Effect, 102 2.5.2.1 Compact CPW Bandpass Filters with Spurious Suppression, 105 2.5.2.2 Compact Microstrip Wideband Bandpass Filters with Ultrawideband Spurious Suppression, 108 References, 114 3 Metamaterial Transmission Lines: Fundamentals, Theory, Circuit Models, and Main Implementations 3.1 Introduction, Terminology, and Scope, 119 3.2 Effective Medium Metamaterials, 122 3.2.1 Wave Propagation in LH Media, 123 3.2.2 Losses and Dispersion in LH Media, 125 3.2.3 Main Electromagnetic Properties of LH Metamaterials, 127 3.2.3.1 Negative Refraction, 128 3.2.3.2 Backward Cerenkov Radiation, 129

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3.2.4 Synthesis of LH Metamaterials, 131 3.2.4.1 Negative Effective Permittivity Media: Wire Media, 132 3.2.4.2 Negative Effective Permeability Media: SRRs, 136 3.2.4.3 Combining SRRs and Metallic Wires: One-Dimensional LH Medium, 139 3.3 Electrically Small Resonators for Metamaterials and Microwave Circuit Design, 141 3.3.1 Metallic Resonators, 142 3.3.1.1 The Non-Bianisotropic SRR (NB-SRR), 142 3.3.1.2 The Broadside-Coupled SRR (BC-SRR), 142 3.3.1.3 The Double-Slit SRR (DS-SRR), 143 3.3.1.4 The Spiral Resonator (SR), 144 3.3.1.5 The Folded SIR, 144 3.3.1.6 The Electric LC Resonator (ELC), 145 3.3.1.7 The Open Split-Ring Resonator (OSRR), 146 3.3.2 Applying Duality: Complementary Resonators, 146 3.3.2.1 Complementary Split-Ring Resonator (CSRR), 147 3.3.2.2 Open Complementary Split-Ring Resonator (OCSRR), 149 3.4 Canonical Models of Metamaterial Transmission Lines, 149 3.4.1 The Dual Transmission Line Concept, 150 3.4.2 The CRLH Transmission Line, 154 3.4.3 Other CRLH Transmission Lines, 158 3.4.3.1 The Dual CRLH (D-CRLH) Transmission Line, 158 3.4.3.2 Higher-Order CRLH and D-CRLH Transmission Lines, 159 3.5 Implementation of Metamaterial Transmission Lines and Lumped-Element Equivalent Circuit Models, 162 3.5.1 CL-Loaded Approach, 162 3.5.2 Resonant-Type Approach, 166 3.5.2.1 Transmission Lines based on SRRs, 167 3.5.2.2 Transmission Lines based on CSRRs, 177 3.5.2.3 Inter-Resonator Coupling: Effects and Modeling, 183 3.5.2.4 Effects of SRR and CSRR Orientation: Mixed Coupling, 191 3.5.2.5 Transmission Lines based on OSRRs and OCSRRs, 195 3.5.2.6 Synthesis Techniques, 203 3.5.3 The Hybrid Approach, 204 References, 206 4 Metamaterial Transmission Lines: RF/Microwave Applications 4.1 Introduction, 214 4.2 Applications of CRLH Transmission Lines, 215 4.2.1 Enhanced Bandwidth Components, 215 4.2.1.1 Principle and Limitations, 215

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4.2.1.2 Illustrative Examples, 219 4.2.2 Dual-Band and Multiband Components, 225 4.2.2.1 Principle for Dual-Band and Multiband Operation, 227 4.2.2.2 Main Approaches for Dual-Band Device Design and Illustrative Examples, 228 4.2.2.3 Quad-Band Devices based on Extended CRLH Transmission Lines, 246 4.2.3 Filters and Diplexers, 250 4.2.3.1 Stopband Filters based on SRR- and CSRR-Loaded Lines, 250 4.2.3.2 Spurious Suppression in Distributed Filters, 251 4.2.3.3 Narrow Band Bandpass Filters and Diplexers Based on Alternate Right-/Left-Handed Unit Cells, 255 4.2.3.4 Compact Bandpass Filters based on the Hybrid Approach, 258 4.2.3.5 Highpass Filters Based on Balanced CRLH Lines, 270 4.2.3.6 Wideband Filters Based on OSRRs and OCSRRs, 270 4.2.3.7 Elliptic Lowpass Filters Based on OCSRRs, 277 4.2.4 Leaky Wave Antennas (LWA), 282 4.2.5 Active Circuits, 290 4.2.5.1 Distributed Amplifiers, 290 4.2.5.2 Dual-Band Recursive Active Filters, 298 4.2.6 Sensors, 300 4.3 Transmission Lines with Metamaterial Loading and Applications, 303 4.3.1 Multiband Planar Antennas, 304 4.3.1.1 Multiband Printed Dipole and Monopole Antennas, 304 4.3.1.2 Dual-Band UHF-RFID Tags, 310 4.3.2 Transmission Lines Loaded with Symmetric Resonators and Applications, 314 4.3.2.1 Symmetry Properties: Working Principle for Sensors and RF Bar Codes, 315 4.3.2.2 Rotation, Displacement, and Alignment Sensors, 316 4.3.2.3 RF Bar Codes, 324 References, 327 5 Reconfigurable, Tunable, and Nonlinear Artificial Transmission Lines 339 5.1 Introduction, 339 5.2 Materials, Components, and Technologies to Implement Tunable Devices, 339 5.2.1 Varactor Diodes, Schottky Diodes, PIN Diodes, and Heterostructure Barrier Varactors, 340 5.2.2 RF-MEMS, 342 5.2.3 Ferroelectric Materials, 344 5.2.4 Liquid Crystals, 346

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5.3 Tunable and Reconfigurable Metamaterial Transmission Lines and Applications, 347 5.3.1 Tunable Resonant-Type Metamaterial Transmission Lines, 347 5.3.1.1 Varactor-Loaded Split Rings and Applications, 347 5.3.1.2 Tunable SRRs and CSRRs Based on RF-MEMS and Applications, 362 5.3.1.3 Metamaterial Transmission Lines Based on Ferroelectric Materials, 375 5.3.2 Tunable CL-Loaded Metamaterial Transmission Lines, 377 5.3.2.1 Tunable Phase Shifters, 378 5.3.2.2 Tunable Leaky Wave Antennas (LWA), 381 5.4 Nonlinear Transmission Lines (NLTLs), 385 5.4.1 Model for Soliton Wave Propagation in NLTLs, 386 5.4.2 Numerical Solutions of the Model, 391 References, 395 6 Other Advanced Transmission Lines

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6.1 Introduction, 402 6.2 Magnetoinductive-wave and Electroinductive-wave Delay Lines, 402 6.2.1 Dispersion Characteristics, 403 6.2.2 Applications: Delay Lines and Time-Domain ReflectometryBased Chipless Tags for RFID, 406 6.3 Balanced Transmission Lines with Common-Mode Suppression, 411 6.3.1 Strategies for Common-Mode Suppression, 411 6.3.1.1 Differential Lines Loaded with Dumbbell-Shaped Slotted Resonators, 412 6.3.1.2 Differential Lines Loaded with CSRRs, 412 6.3.2 CSRR- and DS-CSRR-Based Differential Lines with Common-Mode Suppression: Filter Synthesis and Design, 414 6.3.3 Applications of CSRR and DS-CSRR-Based Differential Lines, 418 6.3.3.1 Differential Line with Common-Mode Suppression, 418 6.3.3.2 Differential Bandpass Filter with Enhanced Common-Mode Rejection, 421 6.3.4 Balanced Filters with Inherent Common-Mode Suppression, 421 6.3.4.1 Balanced Bandpass Filters Based on OSRRs and OCSRRs, 423 6.3.4.2 Balanced Bandpass Filters Based on Mirrored SIRs, 425 6.4 Wideband Artificial Transmission Lines, 429 6.4.1 Lattice Network Transmission Lines, 429 6.4.1.1 Lattice Network Analysis, 430 6.4.1.2 Synthesis of Lattice Network Artificial Transmission Lines, 434 6.4.1.3 The Bridged-T Topology, 437

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6.4.2 Transmission Lines Based on Non-Foster Elements, 439 6.5 Substrate-Integrated Waveguides and Their Application to Metamaterial Transmission Lines, 441 6.5.1 SIWs with Metamaterial Loading and Applications to Filters and Diplexers, 444 6.5.2 CRLH Lines Implemented in SIW Technology and Applications, 445 References, 454 Appendix A. Equivalence between Plane Wave Propagation in Source-Free, Linear, Isotropic, and Homogeneous Media; TEM Wave Propagation in Transmission Lines; and Wave Propagation in Transmission Lines Described by its Distributed Circuit Model 460 Appendix B. The Smith Chart

468

Appendix C. The Scattering Matrix

474

Appendix D. Current Density Distribution in a Conductor

480

Appendix E. Derivation of the Simplified Coupled Mode Equations and Coupling Coefficient from the Distributed Circuit Model of a Transmission Line

482

Appendix F. Averaging the Effective Dielectric Constant in EBG-Based Transmission Lines

484

Appendix G. Parameter Extraction

486

Appendix H. Synthesis of Resonant-Type Metamaterial Transmission Lines by Means of Aggressive Space Mapping

491

Appendix I. Conditions to Obtain All-Pass X-Type and Bridged-T Networks

503

Acronyms

505

Index

508

PREFACE

Transmission lines and waveguides are essential components in radiofrequency (RF) and microwave engineering for the guided transmission of electromagnetic (EM) energy (power and information signals) between two points. Moreover, transmission lines and waveguides are key building blocks for the implementation of passive and active RF/microwave devices of interest in wireless communications (filters, diplexers, splitters, couplers, amplifiers, oscillators, mixers, etc.). In planar technology, lowcost devices can be fabricated by etching patterns (a set of transmission lines and stubs providing certain functionality) in a printed circuit board (PCB), avoiding the use of lumped components, such as capacitors, inductors, or resonators. Transmission linebased circuits are usually designated as distributed circuits, since transmission lines can be described by a network of distributed parameters. In certain designs (e.g., amplifiers and mixers), it is necessary to combine distributed and lumped active elements, such as diodes or transistors. Nevertheless, the main relevant aspect of distributed components is their capability to mimic lumped-reactive elements or a combination of them (e.g., resonators or even more complex reactive circuits). It is thus possible to design fully planar functional devices on the basis of the distributed approach, or to minimize the number of lumped elements (unavoidable in certain components, e.g., active circuits) in the designs. Distributed circuits have two main drawbacks: (1) their dimensions scale with frequency, and (2) transmission lines exhibit very limited design flexibility. Typically, the required transmission lines in the designs have a length of the order of the wavelength, which means that dimensions may be too extreme if the operating frequencies are moderate or low. Concerning the second aspect, distributed circuits are designed by means of transmission lines with certain phase (at the operating frequency) and characteristic impedance. The phase varies linearly with the length of the line and

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PREFACE

frequency (to a first-order approximation transmission lines are dispersionless). Therefore, the functionality of distributed circuits is limited to a certain band; namely, the required nominal phase is lost if frequency deviates from the operating value, which means that distributed circuits are bandwidth limited by nature. The limitations of ordinary transmission lines as building blocks for device design are in part originated from the fact that these lines exhibit a limited number of free parameters for design purposes (by excluding losses, ordinary lines are described by a distributed network with two reactive elements in the unit cell). However, by truncating the uniformity (in the longitudinal direction), by etching patterns in the ground plane, by loading the line with reactive elements, or by using a combination of these (or other) strategies to increase the degrees of freedom of the lines, many possibilities to reduce device size, to improve performance, or to achieve novel functionalities, are open. This book has been mainly conceived to introduce and study alternatives to ordinary lines for the design and implementation of RF/microwave components with superior characteristics in the above cited aspects. We refer to these lines as artificial transmission lines, and the term is as wide as the number of strategies that one can envision to improve the size, performance, or functionality of ordinary lines. The book is devoted to the analysis, study, and applications of artificial lines mostly implemented by means of a planar transmission line (host line) conveniently modified (e.g., with modulation of transverse dimensions, with etched patterns in the metallic layers, and with reactive loading), in order to achieve certain functionality, superior performance, or reduced size. Nevertheless, it will be shown that in certain artificial waveguiding structures, such as electroinductive and magnetoinductive delay lines, the host line is not present. Waveguide-based components are not included in this book, entirely focused on artificial transmission lines in planar technology. Obviously, it is not possible to cover all the material available in the literature, related to the topic of artificial transmission lines, in a single book. Necessarily, the contents of this book are influenced by the personal experience and background of the author. However, many RF/microwave devices and applications of artificial transmission lines reported by other researchers are included in this book, or properly referenced. The book is devoted to readers that are already familiar with RF/microwave engineering. The aim of writing this book has been to provide an up-to-date state of the art in artificial transmission lines, and an in-depth analysis and study of those aspects, structures, devices, and circuits that are more relevant (according to the criterion of the author) for RF/microwave engineering, including design guidelines that can be useful to researchers, engineers, or students involved in the topics covered by this book. Nevertheless, Chapter 1 is dedicated to the fundamentals of planar transmission lines for coherence and completeness, since most of the concepts of this chapter are used in the subsequent chapters, and are fundamental to understand the principles and ideas behind the design and applications of artificial transmission lines. Chapter 2 is focused on artificial transmission lines based on periodic structures, where periodicity plays a fundamental role and is responsible for the presence of band gaps in the transmission spectrum of these lines. The Floquet analysis (leading to the concept of space harmonics), complemented by the coupled mode theory (from which

PREFACE

xv

useful expressions for the design of periodic artificial lines are derived), and the transfer matrix method (useful to obtain the dispersion relation of the fundamental space harmonic), are included in the chapter. The last part is devoted to the applications, which have been divided into those of periodic nonuniform transmission lines (e.g., harmonic and spurious suppression), and those of reactively loaded lines, where not only the reflection properties of periodic structures but also the inherent slowwave effect associated to reactive loading, are exploited. Chapters 3 and 4 are dedicated to artificial transmission lines inspired by metamaterials, or based on metamaterial concepts. The importance of these artificial lines in this book has forced the author to separate the fundamentals/theory and applications into different chapters in order to avoid an excessive chapter length. Thus, Chapter 3 is focused on the theory, circuit models, and main implementations of metamaterial transmission lines, whereas Chapter 4 deals with the applications. Many applications of metamaterial transmission lines are based on the superior controllability of the characteristic impedance and dispersion of these lines, as compared to ordinary lines, related to the presence of reactive elements loading the line. Indeed, metamaterial transmission lines have opened a new way of “thinking” in the design of microwave components, where tailoring the dispersion diagram, and not only the characteristic impedance, is the key aspect (we may accept that metamaterial transmission lines have given rise to microwave circuit design on the basis of impedance and dispersion engineering). The further controllability of the relevant line parameters (phase constant and characteristic impedance) in metamaterial transmission lines, as compared to ordinary lines, has a clear parallelism with the further controllability of the constitutive parameters (permittivity and permeability) in effective media metamaterials (periodic artificial structures exhibiting controllable EM properties, different from those of the materials which they are made). Indeed, we can define an effective permittivity and permeability in metamaterial transmission lines despite that these lines are one-dimensional structures, and we can design the lines in order to support backward (or left-handed) wave propagation (as occurs in metamaterials with simultaneous negative effective permittivity and permeability). However, whereas in effective media metamaterials periodicity and homogeneity (satisfied if the period is much smaller than the wavelength) are necessary conditions to properly define an effective permeability and permittivity, periodicity, and homogeneity are not requirements for impedance and dispersion engineering with metamaterial transmission lines. The former metamaterial transmission lines were implemented by loading a host line with series capacitors and shunt inductors (CL-loaded approach), or by loading the host lines with electrically small resonators, formerly used for the implementation of bulk effective media metamaterials (metamaterial resonators). This latter approach has been called resonant-type approach. Both approaches are included in this book (and many other latter developments), but special emphasis is put on the resonant-type approach. Moreover, in Chapter 4 there are several applications where, rather than the controllability of the impedance and dispersion of the artificial lines, the working principle is the resonance of a transmission line (host line) loaded with metamaterial resonators (these lines are designated as transmission lines with metamaterial loading

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in Chapter 4). Since metamaterial transmission lines are inspired by metamaterials, an introduction to these artificial media and the former implementation are included in Chapter 3. Chapter 3 includes also a section devoted to study the main electrically small resonators useful for the synthesis of metamaterials and microwave circuits based on them (resonant-type approach). In Chapter 4, the applications include enhanced bandwidth components, multiband components, filters and diplexers, active devices with novel functionalities (e.g., distributed amplifiers), novel antennas (e.g., leaky wave antennas and antennas for RFID tags), microwave sensors, and so on. In Chapter 5, the focus is on reconfigurable components based on tunable artificial lines and nonlinear transmission lines. Several materials, components, and technologies (including varactors, RF-MEMS, ferroelectrics, and liquid crystals) for the implementation of tunable components are introduced. Then the chapter focuses on the design of tunable artificial transmission lines and their applications, mostly, although not exclusively, devoted to filters. The last part of the chapter deals with the topic on nonlinear transmission lines, structures that support the propagation of solitons and are of interest for harmonic multiplication. Finally, other advanced transmission lines or, more generally, waveguiding structures are presented and studied in Chapter 6, including applications. The covered topics are electroinductive and magnetoinductive wave delay lines, common-mode suppressed differential lines, lattice network-based transmission lines, transmission lines loaded with non-Foster components, and metamaterial-based substrateintegrated waveguides. Grouping these topics in a single chapter does not obey to a thematic reason, but to the fact that most of them have been recently proposed and/or are still under development, or even to the fact that they are very specific to be included in the previous chapters (e.g., the electroinductive and magnetoinductive wave delay lines and the substrate-integrated waveguides). It is the author’s hope that the present manuscript constitutes a reference book in the topic on artificial transmission lines and their RF and microwave applications, and that the book can be of practical use to researchers, students and engineers involved in RF and microwave engineering, especially to those active in planar circuit and antenna design. FERRAN MARTÍN SANTA MARIA D’OLÓ (BARCELONA) SEPTEMBER 2014

ACKNOWLEDGMENTS

This book is the result of an intensive research activity on the topic of artificial transmission lines carried out by the author and his research group (Centre d’Investigació en Metamaterials per a la Innovació en Tecnologies Electrònica i de les Comunicacions - CIMITEC) at the Universitat Autònoma de Barcelona, and also by many other researchers worldwide, with whom the author has had the privilege to collaborate or interact. It is impossible to include a complete list of all the people that have made possible to write this book. Nevertheless, I must express my most sincere gratitude to several colleagues, friends, and co-workers that have made invaluable contributions to it. I apologize if I omit somebody that deserves to be acknowledged and is not included in the following list. First of all, I would like to give special thanks to the current and past members of my Group (Jordi Bonache, Joan García, Nacho Gil, Marta Gil, Francisco Aznar, Adolfo Vélez, Benito Sans, Gerard Sisó, Ferran Paredes, Gerard Zamora, Miguel Durán-Sindreu, Jordi Selga, Jordi Naqui, Paris Vélez, Simone Zuffanelli, Pau Aguilà, Marco Orellana, Lijuan Su, Marc Sans, Ignasi Cairó, Javier Herraiz, David Bouyge, and Anna Cedenilla), and to the visiting professors (Javier Mata) and students (Kambiz Afrooz and Ali Karami-Horestani). It is an honor to be the head of such productive and fruitful research group (a consequence of the continuous and endless effort of the involved people). Many of the ideas and results presented in the book have their origin in the researchers of CIMITEC, and therefore this book also belongs to them. I would like to highlight Jordi Bonache, who has had many brilliant ideas since more than one decade ago, providing very interesting research results and innovative applications on the basis of artificial transmission lines and related concepts. I must also acknowledge the contribution of Jordi Naqui to this book, who edited many figures of the manuscript; Gerard Zamora, for reviewing part of Chapter 4;

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and Anna Cedenilla, who was in charge of the permissions for the use of many copyrighted figures. I would not like to forget the support of the administrative staff (headed by Mari Carmen Mesas during many years) and technicians (Javier Hellín) of my department, who are not very visible but are essential for the success of the research activities. During the recent years, we have had fruitful collaborations with many groups that have contributed to the progress of the topic covered by the book. Among them, I would like to cite the groups of Prof. Francisco Medina (Universidad de Sevilla); Prof. Mario Sorolla (Universidad Pública de Navarra), who passed away in November 2012; Prof. Vicente Boria (Universitat Politècnica de Valencia); Prof. Rolf Jakoby (TU Darmstadt); Prof. Tatsuo Itoh (University of California Los Angeles); Prof. Christophe Fumeaux (University of Adelaide); Dr. Walter de Raedt (IMEC); Prof. Pierre Blondy (XLIM-Université de Limoges); Prof. Didier Lippens (IEMN-Université de Lille); and the groups involved in the Network of Excellence within the VI Framework Program of the European Union, METAMORPHOSE (2004–2008). We have recently been a partner in the collaborative project Engineering Metamaterials (2008–2014) of the CONSOLIDER INGENIO 2010 Program (MICIIN-Spain), which has represented an ideal platform for cooperation between the partners, and a continuous source of ideas. Special thanks go to Francisco Medina (Universidad de Sevilla), Christian Damm (TU Darmstadt), Silvio Hrabar (University of Zagreb), and Txema Lopetegi (Universidad Pública de Navarra) for reviewing some parts of the manuscript (Txema Lopetegi has also co-authored some sections of Chapter 2 and two appendixes). I will never forget the extraordinary contribution of my past PhD student (shared with Prof. Mario Sorolla), Francisco Falcone (now associate professor at the Universidad Pública de Navarra), to the topic of metamaterial transmission lines (many of the ideas presented in this book are in part due to him). To end the acknowledgments relative to external collaborations, I would like to mention Prof. Ricardo Marqués (Universidad de Sevilla), who is a well-known authority in the field of metamaterials, and with whom I have had the privilege to cooperate, learn, and cowrite a previous book. We have had many stimulating discussions on many topics, including the modeling of artificial lines based on metamaterial concepts. There is no doubt that Prof. Marqués has been a key researcher for the progress and applications of metamaterial transmission lines based on the resonant-type approach. The research activity that has conducted to the results presented in this book has been funded by several agencies or institutions. Particularly, I would like to express my gratitude to the past Spanish Ministry of Science and Innovation (MICIIN), for supporting our work through the collaborative project Engineering Metamaterials, cited earlier (ref. CSD2008-00066), and to the current Spanish Ministry of Economy and Competitiveness (MINECO), for funding our research activities through other national projects. Thanks are also given to the past Spanish Ministry of Industry, Commerce and Tourism (MICyT) for giving us support through several collaborative projects with companies for the development of precompetitive products on the basis of our research activities on topics related to this book. The Government of Catalonia is also acknowledged for giving us support as members of TECNIO (a network of Research and Technology Transfer Centers), and for funding several research projects

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of CIMITEC. I would also like to express my most sincere gratitude to Institució Catalana de Recerca i Estudis Avançats (ICREA) for supporting my work through an ICREA Academia Award (calls 2008 and 2013), and to my university for the continuous support, which includes the Parc de Recerca UAB-Santander Technology Transfer Chair. At the European level, I would like to express my gratitude to the European Commission and to the Eureka Program for funding several international projects. I would also like to mention the Virtual Institute for Artificial Electromagnetic Materials and Metamaterials, “METAMORPHOSE VI AISBL,” for giving us the opportunity to disseminate and promote our research activities, and the Institute of Electrical and Electronics Engineers (IEEE) for elevating me to the grade of IEEE fellow (in acknowledgment to my contributions to the topic of metamaterial-based transmission lines). Last, but not least, I would like to acknowledge the support of my wife, Anna, who has created the necessary atmosphere to write a long manuscript like this, who has accepted my long absences (this extends also to my children, Alba and Arnau), and who has done her best in favor of the family, myself, and my vocation. I would also like to include in the list many other people who have supported or influenced me, such as my parents (Juan and Rosario), my “second parents” (Josep Maria and Josefina), my grandparents, Carlos, Rut, my parents-in-law (Lina and Josep), and many others that are not in my mind at this moment, but are always present in my heart. Thank you very much! FERRAN MARTÍN

1 FUNDAMENTALS OF PLANAR TRANSMISSION LINES

1.1 PLANAR TRANSMISSION LINES, DISTRIBUTED CIRCUITS, AND ARTIFICIAL TRANSMISSION LINES In radiofrequency (RF) and microwave engineering, transmission lines are twoport networks used to transmit signals, or power, between two distant points (the source and the load) in a guided (in contrast to radiated) way. There are many types of transmission lines. Probably, the most well-known transmission line (at least for nonspecialists in RF and microwave engineering) is the coaxial line (Fig. 1.1), which consists of a pair of concentric conductors separated by a dielectric, and is typically used to feed RF/microwave components and to connect them to characterization and test equipment. Other planar transmission lines are depicted in Figure 1.2. There are many textbooks partially or entirely focused on transmission lines and their RF and microwave applications [1–8]. The author recommends these books to those readers interested in the topic of the present book (artificial transmission lines), which are not familiar with conventional (or ordinary) transmission lines. Nevertheless, the fundamentals of planar transmission lines are considered in this chapter for completeness and for better comprehension of the following chapters. As long as waveguides (and even optical fibers) do also carry electromagnetic (EM) waves and EM energy between two points, they can also be considered transmission lines. However, this book is entirely devoted to planar structures; and for this reason, waveguides are out of the scope of this chapter. Artificial Transmission Lines for RF and Microwave Applications, First Edition. Ferran Martín. © 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc.

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FUNDAMENTALS OF PLANAR TRANSMISSION LINES

2b

2a

εr

FIGURE 1.1 Perspective three-dimensional view of a coaxial transmission line. The relevant geometry parameters of the line are indicated, and εr is the relative permittivity (or dielectric constant) of the dielectric material.

Obviously, there are not transmission lines in natural form.1 Transmission lines must be fabricated in order to satisfy certain requirements or specifications; in this sense, they are actually artificial (i.e., man-made) structures. However, the term artificial transmission line is restricted to a specific type of transmission lines, to distinguish them from the conventional ones.2 Before discussing the definition and scope of the term artificial transmission line, let us now point out the different approaches for the study of planar (conventional) transmission lines. If the physical length of the transmission line is much smaller than the wavelength of the transmitted signals, the voltages and currents in the line are uniform, that is, they do not depend on the position in the line.3 Under these conditions, the voltages and currents are dictated by the Kirchhoff’s current and voltage laws and by the terminal equations of the lumped elements present at the input and output ports of the line, or at any position in the line. This is the so-called lumped element approach, which is generally valid up to about 100 MHz, or even further for planar structures (or circuits) including transmission lines not exceeding the typical sizes of printed circuit boards or PCBs (i.e., various centimeters). At higher frequencies, typically above 1 GHz, the finite propagation velocity of the transmitted signals (of the order of the speed of light) gives rise to variations of voltage and current along the lines, and the lumped circuit approach is no longer valid. At this regime, transmission lines can be analyzed by means of field theory, from Maxwell’s equations. However, most planar transmission lines can alternatively be studied and described by means of an intermediate approach between lumped circuits and field equations: the distributed circuit approach. Indeed, for

1

Exceptions to this are, for instance, the axons, which transmit nerve signals in brain neurons. Conventional (or ordinary) transmission lines are uniform along the propagation direction (see Fig. 1.2). 3 Strictly speaking, this is true if losses are negligible. The effects of losses in transmission lines will be discussed later in detail. 2

3

PLANAR TRANSMISSION LINES, DISTRIBUTED CIRCUITS

t

t W

h ε r

Microstrip

CPW

t

t

b

W

εr

S

h ε r

Stripline

Slot line

W

εr

εr

h

t b

G

W

h ε r

W

hl

Inverted suspended microstrip

Suspended microstrip

t

t h ε r

W

S

Coplanar strips

h εr

W

S

Coupled microstrip

t h ε r

W Paired strips

FIGURE 1.2 Perspective three-dimensional view of the indicated planar transmission lines, and relevant geometry parameters. These transmission lines are used for the implementation of distributed circuits, where the shape and transverse dimensions (W, S, G) of the line (or set of lines and stubs) are determined in order to obtain the required line functionality.

4

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

transverse electric and magnetic (TEM),4 or quasi-TEM, wave propagation in planar transmission lines (i.e., the fundamental modes), there is a link between the results inferred from the distributed analysis and field theory. Nevertheless, this connection is discussed and treated in Appendix A, since it is not necessary to understand the contents of the present and the next chapters. The most intriguing aspect of transmission lines operating at microwave frequencies and beyond is the fact that such lines can replace lumped elements, such as capacitances and inductances, in planar circuits, thus avoiding the use of lumped components which increase cost and circuit complexity. Hence, in RF and microwave engineering, transmission lines are not only of interest for signal or power transmission, but they are also key elements for microwave device and component design on the basis of the distributed approach. Thus, the constituent building blocks of distributed circuits are transmission lines and stubs,5 which are implemented by simply etching metallic patterns on a microwave substrate (such patterns define a set of transmission lines and stubs providing certain functionality). Distributed circuits are typically low cost since they are implemented in planar technology. However, the design flexibility, performance, or functionality of planar microwave circuits can be enhanced (and/or their dimensions can be reduced) by loading the lines with reactive elements (not necessarily planar),6 or by breaking the uniformity of the lines in the direction of propagation, or by considering specific arrangements able to provide certain advantages as compared to ordinary lines. In the context of this book, the term artificial transmission line is used to designate these lines with superior characteristics, and to distinguish them from their conventional counterparts (ordinary lines). Hence, notice that the term artificial transmission line is not only restricted to designate artificial structures mimicking the behavior of ordinary lines (e.g., an LC ladder network or a capacitively loaded line acting as a slow wave transmission line).7 In this book, the definition of artificial transmission line is 4 Transmission lines supporting TEM modes require at least two conductors separated by a uniform (homogeneous) dielectric, and the electric and magnetic field lines must be entirely contained in such dielectric. In such modes, the electric and magnetic field components in the direction of propagation are null. A coaxial line is an example of transmission line that supports TEM modes. Microstrip and CPW transmission lines (see Fig. 1.2) are nonhomogeneous open lines, and hence do not support pure TEM modes, but quasiTEM modes. 5 Stubs are short- or open-circuit transmission line sections, shunt or series connected to another transmission line, intended to produce a pure reactance at the attachment point, for the frequency of interest. 6 Notice that this loading refers to line loading along its length, not at the output port (as considered in Section 1.3 in reference to ordinary lines). A line with a load at its output port is usually referred to as terminated line. 7 Artificial lines that mimic the behavior of ordinary lines are sometimes referred to as synthetic lines. Synthetic lines can be implemented by means of lumped, semilumped, and/or distributed components (combination of transmission lines and stubs). Synthetic lines purely based on the distributed approach (e.g., stub-loaded lines) are out of the scope of this book since they are indeed implemented by combining ordinary lines. Other artificial lines that can be considered to belong to the category of synthetic lines (e.g., capacitively loaded lines) are included in this book; but obviously, it is not possible to include all the realizations of synthetic lines reported in the literature. Artificial lines able to provide further functionalities than ordinary lines (e.g., metamaterial transmission lines with multiband functionality) are not considered to be synthetic transmission lines.

5

DISTRIBUTED CIRCUIT ANALYSIS AND MAIN TRANSMISSION LINE PARAMETERS

very broad and roughly covers all those lines that cannot be considered ordinary lines. Nevertheless, in many applications of artificial transmission lines, these lines simply replace ordinary lines, and the design approach of microwave circuits based on such artificial lines is similar to the one for ordinary lines, based on the control of the main line parameters. Therefore, in the next subsections, we will focus the attention on the study and analysis of ordinary lines, including the main transmission line parameters, reflections at the source and load (mismatching), losses in transmission lines, a comparative analysis of the most used planar transmission lines, and examples of applications. Most of these contents will be useful in the following chapters. Other useful contents for this chapter and chapters that follow (and in general for RF/microwave engineering), such as the Smith Chart and the scattering S-matrix, are included for completeness in Appendix B and C, respectively.

1.2 DISTRIBUTED CIRCUIT ANALYSIS AND MAIN TRANSMISSION LINE PARAMETERS Planar transmission lines can be described by cascading the lumped element two-port network unit cell depicted in Figure 1.3, corresponding to an infinitesimal piece of the transmission line of length Δz, and C , L , R , and G are the line capacitance, line inductance, line resistance, and line conductance per unit length, respectively. R is related to conductor losses, whereas G accounts for dielectric losses. From Kirchhoff’s circuit laws applied to the network of Figure 1.3, the following equations are obtained:

v z, t − R Δz i z,t − L Δz

∂i z,t − v z + Δz, t = 0 ∂t

1 1a

∂v z + Δz, t − i z + Δz, t = 0 ∂t

1 1b

i z, t − G Δz v z + Δz, t − C Δz

i(z,t) + v(z,t)

R′Δz

i(z+Δz,t)

L′Δz

G′Δz

–

C′Δz

+ v(z+Δz,t) –

Δz FIGURE 1.3 Lumped element equivalent circuit model (unit cell) of an ordinary transmission line.

6

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

By dividing these equations by Δz, and taking the limit as Δz

0, it follows:

∂v z,t ∂i z,t = − R i z,t − L ∂z ∂t

1 2a

∂i z,t ∂v z, t = − G v z, t − C ∂z ∂t

1 2b

Equations 1.2 are known as the telegrapher equations. If we now consider sinusoidal steady-state conditions (i.e., v(z, t) = V(z) ejωt and i(z, t) = I(z) ejωt), the time variable in the previous equations can be ignored: dV z = − R + jωL I z dz

1 3a

dI z = − G + jωC V z dz

1 3b

and the well-known wave equations result d2 V z − γ2V z = 0 dz2

1 4a

d2 I z − γ2I z = 0 dz2

1 4b

where γ = α + jβ is the complex propagation constant, given by γ=

R + jωL

G + jωC

15

and α and β are the attenuation constant and the phase constant, respectively. Notice that if conductor and dielectric losses can be neglected (R = G = 0), α = 0, and the phase constant is proportional to the angular frequency and given by β=ω L C

16

The general solutions of the wave equations are traveling waves of the form: V z = Vo+e − γz + Vo− e γz

1 7a

I z = Io+e −γz + Io− e γz

1 7b

where the first and second terms correspond to wave propagation in +z and –z directions, respectively. By combining (1.3) and (1.7), it follows that the relation between voltage and current for the traveling waves, also known as the characteristic impedance, is given by

DISTRIBUTED CIRCUIT ANALYSIS AND MAIN TRANSMISSION LINE PARAMETERS

Zo =

Vo+ − Vo− = − = Io+ Io

R + jωL G + jωC

7

18

For lossless lines, the voltage and current in the line are in phase, and the characteristic impedance is a real number: L C

Zo =

19

Although losses may limit the performance of distributed microwave circuits, losses are usually neglected for design purposes, and the propagation constant and characteristic impedance are approximated by (1.6) and (1.9), respectively. According to (1.6), the dispersion relation β−ω is linear. The phase velocity, vp, and the group velocity, vg, are thus identical and given by vp =

vg =

ω = β

dβ dω

1 LC

−1

=

1 LC

1 10

1 11

and the wavelength in the line is given by: λ=

2πvp 2π 2π = = ω β ω LC

1 12

That is, it is inversely proportional to frequency.8 Sometimes, the length of a transmission line (for a certain frequency) is given in terms of the wavelength, or expressed as electrical length, ϕ = βl, where l is the physical length of the line, and ϕ is an angle indicating whether distributed effects should be taken into account or not (as a firstorder approximation, distributed effects are typically neglected if ϕ < π/4). In many distributed circuits, transmission lines and stubs are λ/4 or λ/2 long at the operating frequency, corresponding to electrical lengths of ϕ = π/2 and ϕ = π, respectively. For plane waves in source-free, linear, isotropic, homogeneous, and lossless dielectrics, the wave impedance, defined as the ratio between the electric and magnetic fields, and the phase velocity, are given by [1, 2] (see Appendix A): η=

μ ε

1 13

8 As will be shown, for artificial transmission lines expressions (1.10–1.12) are not necessarily valid. Indeed, for certain artificial lines, the wavelength either increases or decreases with frequency depending on the frequency regions.

8

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

vp =

1 με

1 14

where ε and μ are the dielectric permittivity and magnetic permeability, respectively. These expressions, derived from Maxwell’s equations, do also apply to TEM wave propagation in planar transmission lines, and therefore the main line parameters can be expressed in terms of the material parameters.9 Notice that for nonmagnetic materials μ = μo, the permeability of vacuum, and hence the phase velocity can be rewritten in the usual form: vp =

1 = μo ε

1 c = μo εo εr εr

1 15

where c is the speed of light in vacuum, and ε = εoεr (εo and εr being the permittivity of vacuum and the dielectric constant, respectively). However, for open nonhomogeneous lines, such as microstrip or coplanar waveguide (CPW) transmission lines, where pure TEM wave propagation is not possible, the previous expression does not hold. Nevertheless, the phase velocity in open lines can be expressed as (1.15) by simply replacing the dielectric constant of the substrate material, εr, with an effective dielectric constant, εre, which takes into account the presence of the electric field lines in both the substrate material and air10: vp =

1.3

c εre

1 16

LOADED (TERMINATED) TRANSMISSION LINES

A uniform (in the direction of propagation) transmission line is characterized by the phase constant β (or by the electrical length βl), and by the characteristic impedance, Zo. In a semi-infinitely long transmission line with a traveling wave generated by a source, the characteristic impedance expresses the relation between voltage and current at any transverse plane of the line. If losses are neglected, it follows that the power carried by the traveling wave along the line is given by P +=

9

1 Vo+ 2 Zo

2

1 17

However, the wave impedance should not be confused with the characteristic impedance, Zo, of transmission lines supporting TEM waves, which relates the voltage and current in the line and depends not only on the material parameters but also on the geometry of the line (see Appendix A). 10 See at the end of Appendix A for more details.

9

LOADED (TERMINATED) TRANSMISSION LINES

V+(z) Zo, β

ρL

ρ(z) V–(z)

ZL z=0

FIGURE 1.4 Transmission line terminated with an arbitrary load, located at z = 0.

However, if the line is terminated by a load, three different situations may arise: (1) the incident power is completely absorbed by the load, (2) the incident power is completely reflected by the load, and (3) the incident power is partially absorbed and reflected by the load. Let us consider that the impedance of the load is ZL, that this load is situated in the plane z = 0 of the line (as Fig. 1.4 illustrates), and that a traveling wave of the form V+(z) = Vo+ e−jβz is present in the line. The ratio of voltage to current for such travelling wave is V+(z)/I+(z) = Zo. At z = 0, the relation between the voltage, VL, and the current, IL, in the load must satisfy the Ohm law, that is, VL/IL = ZL. Since, in general, ZL Zo, a reflected wave must be generated at z = 0, so that the Ohm law is preserved. Therefore, the voltage and current in the line can be expressed as follows: V z = Vo+e −jβz + Vo− e jβz

1 18a

Vo+ − jβz Vo− jβz e − e Zo Zo

1 18b

I z =

By forcing the Ohm law at z = 0, it follows that ZL =

V 0 Vo+ + Vo− = + Zo I 0 Vo − Vo−

1 19

and the relation between the amplitude of the reflected and the incident wave, also known as reflection coefficient, is ρL =

Vo− ZL − Zo = Vo+ ZL + Zo

1 20

From (1.20), it follows that if ZL = Zo (matched load), ρL = 0 and the incident power is absorbed by the load (i.e., there are not reflections in the load). Conversely, if the load is an open or a short circuit, the reflection coefficient is ρL(ZL = ∞) = 1 and ρL(ZL = 0) = −1, respectively, and the incident power is reflected back to the source. Notice that the incident power is also reflected back to the source for reactive loads, where |ρL(ZL = jχ)| = 1, χ being the reactance. Partially reflected and absorbed power

10

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

occurs for resistive loads not matched to the line, or for complex loads. Notice also that for passive loads (ZL = R + jχ, with R > 0), the modulus of the reflection coefficient is |ρL| ≤ 1. This is expected since the reflected power, given by P− =

1 Vo+ 2 Zo

2

ρL

2

1 21

cannot be higher than the incident power (given by 1.17) for passive loads. In microwave engineering, the reflection coefficient is typically expressed in dB and identified as the return loss: RL = − 20 log ρL

1 22

For infinitely long transmission lines or for transmission lines terminated with a matched load, constant amplitude travelling waves are present in the line. However, if a reflected wave is generated in the load plane, a standing wave is generated in the line, where the amplitude is modulated by the modulus of the reflection coefficient. From (1.18a) and (1.20), the voltage in the line can be written as follows: V z = Vo+ e − jβz + ρL e jβz

1 23

If we now express the reflection coefficient in polar form (ρL = |ρL| ejθ), the voltage in the line can be rewritten as follows: V z = Vo+e − jβz 1 + ρL e2jβz + jθ

1 24

from which it follows: V z

2

= Vo+

2

1 + ρL 2 + 2 ρL cos 2βz + θ

1 25

Equation 1.25 indicates that the amplitude is a maximum (Vmax = |Vo+|[1+|ρL|]) and a minimum (Vmin = |Vo+|[1−|ρL|]) at planes separated by λ/4, and the ratio between the maximum and minimum voltage in the line, known as voltage standing wave ratio, is given by SWR =

1 + ρL 1 − ρL

1 26

As anticipated, the SWR is determined by the reflection coefficient. However, it only depends on the modulus of the reflection coefficient, not on its phase, θ. This means that from the information of the SWR, it is not possible to completely characterize the load. For instance, it is not possible to distinguish between a short circuit, an open circuit, or a reactive load, since the reflection coefficient of these loads has the same modulus (|ρL| = 1). Nonetheless, in many applications the relevant information is the

11

LOADED (TERMINATED) TRANSMISSION LINES

matching between the line (or the source) and the load in terms of the power transmitted to the load, the phase information being irrelevant. Although wave reflection in a transmission line is caused by a mismatch between the line and the load, and hence it is ultimately generated at the plane of the load (z = 0), the reflection coefficient can be generalized to any plane of the line, as the ratio between the voltage of the incident and reflected wave, that is ρz =

V− V − e jβz = +o − jβz = ρL e2jβz + V Vo e

1 27

where, as expected, |ρ(z)| = |ρL|. One important point of terminated lines is the amount of power delivered by a given source to the line. If the source has complex impedance, Zs, such power is directly characterized by the power wave reflection coefficient, s, given by [9]: s=

Zin − Zs∗ Zin + Zs

1 28

where the asterisk denotes complex conjugate, and Zin is the impedance seen from the input port of the line, that is, looking into the load (Fig. 1.5). Actually, the power transmission coefficient, which is the relevant parameter for computing the power transmitted to the line for the general situation of a source with complex impedance, is given by τ = 1− s 2 = 1−

Zin − Zs∗ Zin + Zs

2

1 29

The input impedance, Zin, depends on the distance between the load and the input port (i.e., the plane of the source). This impedance can be simply computed as follows: Zin =

V − l Vo+ e jβl + ρL e − jβl = + jβl Zo I −l Vo e − ρL e − jβl

1 30

Zs + Vs –

Zin z=–l

Zo, β

ZL z=0

FIGURE 1.5 Transmission line of length l fed by a voltage source and terminated with an arbitrary load, located at z = 0.

12

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

And after some minor manipulation, Zin = Zo

ZL + jZo tan βl Zo + jZL tan βl

1 31

The analysis of (1.31) reveals that in the limit βl 0, Zin = ZL, as expected, since this regime corresponds to the lumped element approximation discussed in Section 1.1. Expression (1.31) also indicates that the need to model the line as a distributed circuit does not solely depend on frequency (through β), but also on the line length l. Indeed, the key parameter is the electrical length, βl, as anticipated in the previous subsection. From (1.31), it follows that the input impedance is a periodic function with βl, and hence a periodic function with both the line length and the frequency. If the frequency is set to a certain value, the input impedance is a periodic function of period λ/2 with the line length. This means that the input impedance looking into the load seen from planes separated by a multiple of λ/2 is identical. From this result, it follows that for a λ/2 line (βl = π), the input impedance is the one of the load, Zin = ZL. Let us now consider several cases of particular interest. If the line length is l = λ/4 (βl = π/2), the input impedance is Zin =

Zo2 ZL

1 32

which means that the input impedance is inversely proportional to the load impedance, and hence a λ/4 transmission line acts as an impedance inverter. This means that a reactive load with inductive/capacitive reactance is seen as capacitive/inductive reactance from the input port; in other words, the sign of the reactance is reversed in λ/4 lines. From (1.32), it also follows that an open-circuit load is transformed to shortcircuit at the input port of a λ/4 line, and vice versa. For the general case of open-ended (ZL = ∞) and short-circuited (ZL = 0) lines, the input impedance given by (1.31) takes the following form: Zin ZL = ∞ = − jZo cot βl

1 33a

Zin ZL = 0 = jZo tan βl

1 33b

Thus, the input impedances are purely reactive, just as those of lumped reactive elements (inductors and capacitors). For lines satisfying βl < π/2, a short-circuited line resembles an inductor, whereas an open-ended line mimics a capacitor. However, the reactances of lumped inductors and capacitors have different mathematical forms than those of shorted and opened transmission lines. This means that we cannot replace lumped reactive elements with open or shorted lines exhibiting identical behavior. However, by forcing the impedance of a capacitor and inductor to be equal to those of (1.33) we obtain

13

LOADED (TERMINATED) TRANSMISSION LINES

−

j = − jZo cot βl ωC

1 34a

jωL = jZo tan βl

1 34b

and the previous expressions have solutions at many different frequencies. Let us consider the smallest of these angular frequencies and call it ωc. If we set the length of the line to be λ/8 at this frequency (i.e., βl = π/4 at ωc), the previous expressions take the following form: 1 = Zo ωc C

1 35a

ωc L = Zo

1 35b

Therefore, if we wish to implement a short-circuited transmission line with the same impedance as an inductor L at frequency ωc, we set the characteristic impedance of the transmission line to Zo = ωcL. Likewise, if we wish to obtain the reactance of a capacitor C at frequency ωc, we set the characteristic impedance of the transmission line to Zo = 1/ωcC. In both cases, the length of the line must be set to λ/8 at ωc. Expressions (1.35) are called Richard’s transformations [10], and are useful to avoid the use of lumped reactive components in certain microwave circuits such as low-pass filters. However, since Richard’s transformations guarantee identical reactances between the lumped and the distributed reactive elements at a single frequency, we cannot expect that the response of a lumped circuit is identical to that of the distributed counterpart. Obviously, the load of a transmission line can be another transmission line with different characteristic impedance. Let us consider that two transmission lines of characteristic impedances Zo and Z1, respectively, are cascaded as shown in Figure 1.6, and that the transmission line to the right of the contact plane (z = 0), that is, the line acting as load, is either infinitely long or terminated with a matched load (so that there are not reflected waves in this line). Under these conditions, the load impedance seen by the transmission line to the left of z = 0 is simply Z1. Hence, the reflection coefficient at z = 0 is given by ρ=

V+(z) Zo

Z1 − Zo Z1 + Zo

ρ

1 36

V+(z) Z1

V–(z) z=0 FIGURE 1.6 Cascade connection of two transmission lines with different characteristic impedance.

14

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

and the incident wave is partially transmitted to the second line. For z < 0, the voltage in the line can be expressed as (1.23), whereas to the right of the contact plane, the voltage can be expressed in terms of a transmission coefficient, T, as follows: V z = Vo+Te − jβz

1 37

By forcing expressions (1.23) and (1.37) to be identical at z = 0, the transmission coefficient is found to be T =1+ρ=

2Z1 Z1 + Zo

1 38

and the transmission coefficient expressed in dB is identified as the insertion loss: IL = − 20 log T

1 39

Notice that the transmission coefficient, defined by the fraction of the amplitude of the voltage of the incident wave transmitted to the second transmission line (expression 1.37), can be higher than one (this occurs if Z1 > Zo). This result does not contradict any fundamental principle (i.e., the conservation of energy), since the transmitted power is always equal or less than the incident power (however, the amplitude of the voltage of the transmitted wave can be higher than Vo+). To end this subsection, let us briefly consider the reflections generated by a source with mismatched impedance that feeds a transmission line with characteristic impedance Zo and length l (Fig. 1.5). Let us assume that the load of the transmission line is also mismatched, so that a reflected wave is generated by the load once the incident wave reaches the load plane. Once the switch is closed at t = 0, the following expression must be satisfied at any time t > 0: Vs = Zs I − l + V − l

1 40

Before the reflected wave at the load reaches the plane of the source (z = −l), that is, for t < 2l/vp, expression (1.40) is written as follows: Vs = Z s

V1+ + V1+ Zo

1 41

where V1+ is the amplitude of the incident wave generated by the source after the switch is closed, that is,11

11

This expression is valid if the source impedance is purely resistive. However, expression (1.45) is valid for any source impedance (purely resistive, purely reactive, or complex). The reason is that the time domain analysis giving (1.45) can be initiated once the transient associated to reactive or complex load and/or source impedances has expired.

15

LOADED (TERMINATED) TRANSMISSION LINES

V1+ = Vs

Zo Zo + Zs

1 42

Once the reflected wave V1− reaches the source plane, a reflected wave V2+ must be generated by the source in order to satisfy the Ohm law. Hence, (1.40) is expressed as follows: Vs = Zs

V1+ − V1− + V2+ + V1+ + V 1− + V2+ Zo

1 43

By combining (1.41) and (1.43), we obtain the following: Zs − V − V2+ = V1− + V2+ Zo 1

1 44

and the reflection coefficient at the source is found to be ρs =

V2+ Zs − Zo = V1− Zs + Zo

1 45

which is formally identical to the reflection coefficient at the load. Obviously, when the reflected wave at the source (V2+) reaches the load plane, a new wave (V2−) is generated by reflection at the load plane, and the process continues indefinitely until the steady state is achieved. This endless bouncing process converges to a steady state since the amplitude of the reflected waves progressively decreases.12 Let us calculate, as an illustrative example, the steady-state voltage at z = −l for the structure of Figure 1.5. If the initial wave is designated as V1+, the first reflected one at z = −l, taking into account the phase shift experienced by the wave along the transmission line, is V1− = V1+ρL e−2jβl (see Fig. 1.7). The next two are V2+ = V1+ρ L ρs e−2jβl and V2− = V1+ρL2ρs e−4jβl. The steady-state voltage at z = −l is given by the superposition of the left to right (+) and right to left (−) waves, that is13, Vi + +

V z = −l = i

Vi− = V1+ i

1 ρi + V1+ 1 − ρi ρs 1 − ρi ρs

1 46

where V1+ is given by (1.42) and ρi = ρL e−2jβl. Introducing (1.42) and (1.45) in (1.46) gives V z = − l = Vs

12

Zo 1 + ρL e − 2jβl Zo + Zs − Zs − Zo ρL e −2jβl

1 47

This is consequence of the modulus of the reflection coefficients at the source and the load, which is smaller than one for passive loads. However, if the line is loaded with an active load, instability is potentially possible. 13 To derive (1.46), the identity 1 + x + x2 + … = 1/(1 − x), where |x| < 1, has been used.

16

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

V1+ V1 + e – j β l l/vp ρL V1+ e – j β l

ρL V1+ e – 2 j β l 2l/vp ρS ρL V1+ e – 2 j β l

ρS ρL V1+ e – 3 j β l

3l/vp ρS ρ L2 V1+ e – 4 j β l

ρSρL2 V1+ e – 3 j β l

4l/vp

t

ρS2 ρL2 V1+ e – 4 j β l z=0

z = –l

FIGURE 1.7 Bounce diagram corresponding to the example discussed in the text. The vertical axis is the time axis.

which can be expressed as follows: V z = − l = Vs

Zin Zin + Zs

1 48

where Zin, given by (1.30), is the impedance seen from the source plane. As expected, the steady-state voltage at z = −l is simply given by the voltage divider, considering the series connection of the source impedance and the input impedance of the loaded line.

1.4

LOSSY TRANSMISSION LINES

In planar transmission lines of practical interest for RF and microwave circuit design losses are small and they are usually neglected for design purposes (as it was mentioned in Section 1.2).14 However, although small, losses produce 14

To guarantee small losses, distributed circuits must be preferably implemented in commercially available low-loss microwave substrates.

17

LOSSY TRANSMISSION LINES

attenuation and distortion in the transmitted signals, and the analysis of their effects on wave propagation is of interest. There are three main causes of losses: (1) the finite conductivity of the metals (conductor losses), (2) the dissipation in the dielectric (either caused by the presence of free electrons or by dipole relaxation phenomena), and (3) radiation losses. Although radiation losses may be dominant under some circumstances, transmission lines operating as guided-wave structures must be designed in order to exhibit small radiation. Hence, this loss mechanism is not considered by the moment.15 Ohmic (or conductor) and dielectric losses are accounted for by the lumped element circuit model of the transmission line (Fig. 1.3) through the series resistance R and shunt conductance G , respectively. Let us now calculate the complex propagation constant (expression 1.5) under the low-loss approximation (justified by the reasons explained earlier), namely, R ωL and G ωC . The complex propagation constant can be rearranged and written as follows:

γ = jω L C

1−j

R G + ωL ωC

−

RG ω2 L C

1 49

Neglecting the last term in the square root of (1.49) and applying the Taylor series expansion up to the first order, the complex propagation constant can be approximated by γ = jω L C 1 −

j R G + 2 ωL ωC

1 50

From (1.50), the attenuation constant and the phase constant for low-loss transmission lines can be easily identified:

α=

1 R + G Zo 2 Zo

1 51

β=ω L C

1 52

where Zo is the lossless characteristic impedance of the line given by (1.9).

15

Nevertheless, leaky-wave transmission lines are specifically designed to enhance radiation. These lines are the building blocks of leaky-wave antennas (LWAs), as will be shown in Chapter 4.

18

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

With regard to the characteristic impedance (expression 1.8), it can be rearranged and written as follows:

Zo =

L C

Cω Lω − 1−j G 2 R LCω Lω −j R RG

1 53

By virtue of the low-loss approximation, the second term of the denominator in the square root can be neglected, and, by using the first order Taylor series approximation, the characteristic impedance is found to be Zo =

L j R G − 1− 2 ωL ωC C

1 54

and this expression can be further simplified to (1.9).16 According to these results, it follows that the phase constant and the characteristic impedance of low-loss transmission lines can be closely approximated by considering the line as lossless. The attenuation constant (1.51) has two contributions: one associated to conductor losses (proportional to R ), and one associated to dielectric losses (proportional to G ). Despite for low-loss lines the phase constant can be approximated by a linear function, the effects of dispersion may be appreciable in very long transmission lines, and may give rise to signal distortion. However, there is a special case where the phase constant of lossy transmission lines varies linearly with frequency, and dispersion is not present. Such lines are called distortionless lines and must satisfy the following identity (known as distortionless, or Heaviside, condition) R C =G L

1 55

In view of the general expression of the complex propagation constant (1.5), the unique way to achieve a linear dependence of β with frequency for a lossy line is to achieve a complex propagation constant of the form A + jωB (where A and B are constants). The only way this can be satisfied is if R + jωL and G + jωC differ by no more than a constant factor. This means that both the real and imaginary parts must be independently related by the same factor, which leads to (1.55). Under the condition specified by (1.55), the complex propagation constant is17 γ= 16

R + jω L C Zo

1 56

Notice that this means to neglect R /ωL and G /ωC in (1.54). However, if these two terms are neglected in (1.50), we find the trivial solution corresponding to the lossless line, with β given by (1.6) and α = 0. 17 Notice that, using (1.55), the attenuation constant can also be expressed as α = (R G )1/2 or α = G Zo.

19

LOSSY TRANSMISSION LINES

and β is given by (1.6). It is interesting to mention that the attenuation constant (α = R /Zo) does not depend on frequency, which means that all frequency components are attenuated the same factor. This means that distortionless lines are able to transmit pulse signals or modulated signals without distortion (although these signals are attenuated along the line due to losses). It is also simply to demonstrate that the characteristic impedance of distortionless lines is a real constant given by (1.9). In practical transmission lines, the distortionless condition is not easy to satisfy since G is usually very small. To compensate this, the line can be loaded with series connected inductances periodically spaced along the line. This strategy leads to an unconventional transmission line that can indeed be considered an artificial transmission line.18 Nevertheless, the elements of the distributed circuit of a transmission line are not exactly constant (in particular, R varies weakly with frequency), and the distortionless condition (expression 1.55) is difficult to meet in practice. For planar transmission lines used as building blocks in distributed circuits, where the lines are low-loss and short, dispersion is not usually an issue, except at high frequencies (dozens of GHz) or for very wideband signals. 1.4.1 Dielectric Losses: The Loss Tangent Although for the design of most planar distributed circuits losses are neglected in the first steps, it is important to simulate their effects before fabrication. In EM solvers, losses are introduced by providing the conductivity of the metallic layers, and the loss tangent (tanδ) of the substrate material. The loss tangent takes into account dielectric losses, including both conduction losses (due to nonzero conductivity of the material) and losses due to damping of the dipole moments (that represents an energy transfer between the external electric field and the material at microscopic level). For a dielectric material, the application of an electric field gives rise to the polarization of the atoms or molecules of the medium in the form of electric dipole moments, which contribute to the total displacement flux according to D = εo E + Pe

1 57

where Pe is the electric polarization, which is related to the applied electric field through19 Pe = εo χ e E 18

1 58

According to the definition of artificial transmission line adopted in this book (see Section 1.1), L-loaded distortionless lines belong to this group, but such lines are out of the scope of this manuscript. As will be seen in Chapter 2, periodic loaded lines exhibit a cut-off frequency. Beyond this frequency, attenuation dramatically increases, and hence these lines may not support the transmission of high-frequency or broadband signals. 19 It is assumed that the material is linear, isotropic, and homogeneous, that is, the electric susceptibility and permittivity are scalars that do not depend on the position and magnitude of the external field. For anisotropic materials, the relation between E and Pe , or between E and D, is a tensor.

20

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

χ e being the electric susceptibility. Combining the previous expressions, the electric displacement can be written as follows: D = εo 1 + χ e E = ε E

1 59

In vacuum, the electric susceptibility is null, whereas in dielectric materials it is a complex number, where the imaginary part accounts for losses. In low-loss materials, the imaginary part of the susceptibility can be neglected to a first-order approximation. However, the effects of material losses on circuit performance may play a role in distributed circuits.20 For this reason, losses cannot be neglected in the evaluation of circuit performance (typically inferred from commercially available EM simulators). In lossy materials, the dielectric permittivity is thus a complex number that can be expressed as follows: ε = εo 1 + χ e = ε − jε

1 60

where ε is a positive number due to energy conservation [1, 11]. Conduction losses, associated to the presence of free electrons in the dielectric material (it is assumed that the material exhibits a nonzero conductivity), do also contribute to the imaginary part of the complex permittivity. The conduction current density is related to the electric field through the Ohm law: J =σE

1 61

where σ is the conductivity of the material. By introducing the previous expression in the Ampere–Maxwell law, we obtain ∇ × H = jωε E + σ E ∇ × H = jω ε − j

σ + ωε ω

1 62a E

1 62b

From (1.62), it is clear that the effects of conduction losses can be accounted for by including a conductivity dependent term in the imaginary part of the complex permittivity. The loss tangent is defined as the ratio between the imaginary and real parts of this generalized21 permittivity: 20

In low-loss microwave substrates, losses are dominated by the finite conductivity of the metal layers. However, in general-purpose substrates, such as FR4, or in high-resistivity silicon (HR-Si) substrates, among others, material losses may significantly degrade the circuit performance. 21 The complex permittivity including the contribution of σ is usually designated as effective complex permittivity in most textbooks. However, in this book, the term effective permittivity is either referred to the permittivity of effective media, or metamaterials, as will be seen in Chapter 3, or it is used to describe wave propagation in quasi-TEM lines by introducing an “averaged” permittivity (and permeability) in the equations governing purely TEM wave propagation (see Appendix A).

21

LOSSY TRANSMISSION LINES

Magnetic walls

ε, σ

h W

FIGURE 1.8 Parallel plate transmission line with magnetic walls at the edges.

tan δ =

σ + ωε ωε

1 63

Low-loss dielectrics are characterized by a small tanδ (typically 10–4 − 10–2). Let us now try to link the loss tangent to the dielectric contribution of the attenuation constant for low-loss transmission lines. Let us consider a hypothetical transmission line with purely TEM wave propagation, for example, a stripline, or a parallel plate transmission line with magnetic walls at the lateral sides (or with very wide plates to neglect the fringing fields). This simplifies the analysis, and provides compact formulas, which is enough for our purposes. For the case of the parallel plate transmission line (Fig. 1.8), let W and h be the width of the metal plates and the height of the substrate, respectively. The per-unit-length line conductance is related to the dielectric conductivity as22 G =

W σ h

1 64

which in turn can be expressed in terms of the per unit length capacitance as G =

22

C σ ε

1 65

We assume that the conductivity in (1.64) is actually the effective conductivity, given by the numerator of (1.63).

22

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

The dielectric contribution of the attenuation constant for low-loss lines is given by the second term of the right-hand side of (1.51), and can be written as follows: 1 αd = G 2

L C

1 66

Introducing (1.65) in (1.66), we finally obtain

αd =

1C σ 2ε

L 1σ β β 1 = tan δ = ω = 2 C 2ε ω 2

με tan δ

1 67

In nonhomogeneous open lines, such as microstrip lines or CPWs, expression (1.67) is not strictly valid, but it provides a rough approximation of αd by merely introducing the effective permittivity (defined as the effective dielectric constant times the permittivity of vacuum) in the last term. Indeed, for microstrip transmission lines (1.67) rewrites as [12, 13] follows: αd =

πf εr εre − 1 tan δ c εre εr − 1

1 68

where it has been assumed that μ = μo and (μoεo)−1 is the speed of light in vacuum, c. Notice that if εr = εre, (1.67) and (1.68) are identical. The analysis of (1.68) also reveals that for high values of εr (and hence εre), expression (1.67) provides a good estimation of αd by introducing in the root the effective permittivity. The dielectric material (substrate) used in planar transmission lines is characterized by the loss tangent (which accounts for dielectric losses) and by the dielectric constant (which determines the phase velocity). Although these parameters are supplied by the manufacturer with the corresponding tolerances, there are sometimes substantial variations that make necessary the characterization of the material (i.e., the measurement of the dielectric constant and the loss tangent) for an accurate design. The dielectric constant and the loss tangent of a substrate material can be experimentally inferred by means of a microstrip ring resonator configuration (Fig. 1.9) [13–15]. The transmission coefficient exhibits transmission peaks at frequencies that depend on the dielectric constant of the substrate, and the loss tangent is extracted from the quality factor of the resonance peaks along with the theoretical calculations of the conductor losses (see next subsection). For a ring resonator, resonances occur at those frequencies where the ring circumference is a multiple of the wavelength, λ. The resonance condition can thus be expressed as follows: λ=

2πrm vp c 1 = = n εre fn fn

1 69

23

LOSSY TRANSMISSION LINES

(a)

Width (w)

Width (w)

Radius (rm)

Microstrip length (L) Gap (G)

(b) –40

S21 (dB)

–50

–60

–70 Measurement Simulation –80 0.5

1

1.5 Frequency (GHz)

2

2.5

FIGURE 1.9 Microstrip ring resonator configuration used to extract the dielectric constant and loss tangent of the substrate (a), and typical frequency response with transmission peaks (b). Reprinted with permission from Ref. [15]; copyright 2007 IEEE.

where n refers to the nth-order resonance, and rm is the mean ring radius. The effective dielectric constant is thus [13–15] εre =

nc 2πrm fn

2

1 70

Once the effective dielectric constant is known (the resonance frequencies can be easily inferred from the measured transmission coefficient), the dielectric constant is given by [6, 13]:

24

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

2εre + M − 1 M+1

εr =

1 71

where M = 1 + 12

−1 2

h W

1 72

and W is the effective strip width, given by 1 25t 2h 1 + ln π t

W =W +

1 73

In (1.72) and (1.73), h and t are the thickness of the substrate and metal layer, respectively, and W is the strip width. Expressions (1.71–1.73) are valid under the assumption that W ≥ h >> t, which is usually satisfied (general expressions are given in [6]). For the determination of tanδ, it is first necessary to measure the unloaded quality factor, given by [13, 14]: Qo =

QL 1 − 10 − IL

20

1 74

where IL is the measured insertion loss at resonance, and the loaded quality factor is given by QL =

fo BW − 3

dB

1 75

BW−3 dB being the −3 dB bandwidth, which can be easily measured from the transmission coefficient. The total attenuation in the resonator is related to the unloaded quality factor by α=

π Qo λ

1 76

where λ is given by (1.69). By subtracting to (1.76) the conduction attenuation constant, given by expression (1.90) (see the next subsection), the dielectric attenuation constant, αd, can be inferred,23 and by using (1.68), the loss tangent can be finally obtained. 23

It is assumed that radiation from the ring is negligible (valid at moderate frequencies) [13]; hence, the contribution of radiation loss to the attenuation constant is null.

25

LOSSY TRANSMISSION LINES

Alternatively, the dielectric constant and the loss tangent of thin film un-clad substrates and low-loss sheet materials can be measured by means of specific instrumentation, namely, a split cylinder resonator. It is a cylindrical resonant cavity separated into two halves, one of them being movable in order to accommodate varying sample thicknesses (i.e., the sample is loaded in the gap between the two cylinder halves). Each cylinder half accommodates a small coupling loop, introduced through a small hole, in order to measure the transmission coefficient of the fundamental TE011 mode. The principle for the determination of the dielectric constant and the loss tangent is the variation of the resonance frequency and quality factor with loaded and un-loaded cylinder (obviously, the thickness of the sample must be accurately known for a correct measurement). More details on this method are given in Refs. [16, 17].

1.4.2 Conductor Losses: The Skin Depth Let us consider a conductor material with finite, but high, conductivity (i.e., a low-loss conductor), where the conduction current dominates over the displacement current, or σ >> |ωε |,|ωε |. The complex propagation constant in such medium, with general expression given by (see Appendix A)

γ = α + jβ = jω

μ ε −j

σ + ωε ω

1 77

can be written as follows: γ = jω

σ 1+j = ω δp

1 78

2 ωμσ

1 79

μ −j

with

δp =

From (1.78), it follows that both the phase constant and the attenuation constant are given by α=β=

1 δp

1 80

Since the attenuation of the fields is given by e−αz = e−z/δp, where z is the direction of propagation, δp indicates the distance over which a plane wave is attenuated by a factor of e−1. If a plane wave in vacuum impinges on a conductor material with its surface perpendicular to the wave vector, the wave transmitted to the conductor decays

26

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

δp r σ l FIGURE 1.10 Cylindrical conductor with conductivity σ. The effective cross section for the calculation of the AC resistance is given by the annular gray region corresponding to one skin depth.

exponentially from its surface. Therefore, δp is referred to as penetration depth.24 It is worth mentioning that (i) δp decreases with the square root of the conductivity and frequency, and (ii) the wavelength is given by λ = 2πδp and is typically much smaller than the wavelength in free space. Similarly, if a conductor such as a wire or a metallic strip is carrying an AC current, the current tends to concentrate in the conductor surface as frequency increases. This phenomenon is known as skin effect and increases the high frequency AC resistance of the conductor since the effective conductor cross section is reduced. To a first-order approximation, the effective cross section of the conductor is limited by the external surface contour and by the curve resulting by reducing such contour by the penetration depth, also known as skin depth. Let us consider as a simple illustrative example a cylindrical conductor with length l and radius r (Fig. 1.10). The AC resistance is given by RAC =

l l = σAAC σ2πrδp

1 81

where AAC is the effective conductor cross section. Expression (1.81) can also be derived by considering the intrinsic impedance of the low-loss conductor, given by25 η=

μ = ε

μ j ω = 1 + j Rs σ

1 82

where Rs is the surface resistance Rs =

24

1 = σδp

μω 2σ

1 83

In copper, the most used clad metal for PCB technology, the penetration depth (or skin depth) at 1 and 10 GHz is 2.06 and 0.66 μm, respectively. 25 Notice that in a good conductor, the phase of the magnetic component of an EM wave propagating through it lags that of the electric component by π/4.

27

LOSSY TRANSMISSION LINES

Considering again a plane wave impinging on the surface of a low-loss conductor, the conduction current density within the conductor is given by26 J = σ E = σEo e

− δzp

e

− jδzp

x

1 84

where Eo is the electric field in the surface, and it has been assumed that the electric field is polarized in the x-direction. The current across a surface normal to x, of width a and infinitely long in the direction normal to the surface (z-direction) is given by: yo +a

∞

I=

dy σEo e

dz 0

− δzp

e

− jδzp

1 85

yo

which gives I = aσEo

δp aEo = 1 + j Rs 1 + j

1 86

The voltage drop between two points separated a distance l in the x-direction is simply V = Eol. Thus, the surface impedance is given by Z s = Rs 1 + j

l a

1 87

which reduces Rs(1 + j) for a square geometry. The term Rs is usually designated as square resistance. For the cylindrical conductor considered earlier, the AC resistance can thus be inferred from the real part of (1.87) by considering a = 2πr, which gives (1.81). Deriving the relation between the conduction loss attenuation constant, αc, and the surface (or square) resistance in most planar transmission lines is not straightforward. Nevertheless, for a parallel plate transmission line, the per-unit-length resistance is roughly given by [12] R =

2Rs W

1 88

where W is the width of the strips. The previous expression is valid under the assumption that the current density is uniform in the transverse plane and concentrated one

26

This result is also obtained from Maxwell’s equations by calculating the current density distribution in the direction normal to the surface (z-direction) of a semi-infinite conductor, with current density parallel to such surface, and assuming that the current density at the surface is Jx (z = 0) = σEo (see Appendix D).

28

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

skin depth from the interface between the conductors and the substrate. Introducing (1.88) in the first term of (1.51), gives αc =

Rs Zo W

1 89

For microstrip lines, closed-form expressions for αc have been derived in [12]. Specifically, if W ≥ 2h27 αc =

Rs 1 Zo h W 2 W + ln 2πe + 0 94 h π 2h × 1+

2

W W πh + h W + 0 94 2h

1 90

h h 2h 1+t h +1 − + ln W πW t 1 + t 2h

The conduction attenuation constants for microstrip lines satisfying W/h ≤ l/2π and l/2π < W/h ≤ 2 are reported in Ref. [12]. It is worth mentioning that (1.90) simplifies to (1.89) in the limit of wide strips (W/h >> 1).

1.5

COMPARATIVE ANALYSIS OF PLANAR TRANSMISSION LINES

The objective of this subsection is to briefly highlight some advantages and limitations of planar transmission lines from a comparative viewpoint. The most used transmission lines for the implementation of planar distributed circuits are microstrip lines and CPWs because no more than two metal levels are needed for their implementation. Striplines are closed and shielded structures, but they require three metal levels and their use is very limited. However, striplines support TEM wave propagation, and the phase velocity in these lines does not depend on their lateral geometry (i.e., the strip width).28 Conversely, microstrip lines and CPWs are nonhomogeneous open lines that do not support purely TEM waves, but quasi-TEM waves. The dielectric substrate slows the waves down, as compared to the field lines in air, and the field lines tend to bend forward, thus preventing the presence of TEM modes. 27

If surface roughness is not negligible, its effects can be accounted for in (1.90) by merely including an additional term in Rs, as reported in [13]. In [12], αc appears multiplied by the factor 8.68 since the units are given in dB/unit length. This factor arises from the conversion from Nepers (Np) to dB. The Neper is a unit in logarithmic scale that uses the natural logarithm. Thus, if an arbitrary variable F(z) is attenuated with position as F(z) = Fo exp(−αz), the loss factor expressed in Np is −ln[F(z)/Fo] = αz, and α is said to have units of Np/unit length. The loss factor expressed in dB is −20log[F(z)/Fo] = −20log[exp(−αz)] = αz20log(e) = 8.68 αz. Hence, the conversion from Np/unit length to dB/unit length introduces the above number 8.68 in expression (1.90). 28 For TEM wave propagation in a stripline, the substrate material must be homogeneous and isotropic. Strictly speaking, purely TEM waves do also require perfect conductors. This latter condition cannot be satisfied in practice, but as long as the resistivity of the conductors is low, purely TEM wave propagation can be assumed.

COMPARATIVE ANALYSIS OF PLANAR TRANSMISSION LINES

29

From the viewpoint of shielding, the advantage of microstrip lines over CPWs is the presence of the ground plane in the back substrate side, which effectively isolates the structure from the backside region, and prevents it from potential interference effects caused by other circuits or materials (including metallic holders). CPWs with backside metallization (also known as conductor-backed CPWs) can also be implemented, but this backside metallization may induce leaky wave propagation as a result of the parasitic parallel plate waveguides present at both sides of the CPW axis. As compared to microstrip lines, CPWs without backside ground plane only need a metallic layer for their implementation. This eases fabrication and the shunt connection of lumped elements, since the ground plane is coplanar to the central (conductor) strip, and vias are not necessary. Nevertheless, in asymmetric CPW structures, such as bended lines, or asymmetrically loaded (along its length) lines, or in CPW lines with discontinuities, the parasitic slot mode29 may be generated and obscure the fundamental mode (and hence degrade device performance). To prevent the presence of the slot mode in asymmetric CPW transmission lines, the ground plane regions must be electrically connected through air bridges, or by means of backside strips and vias (this technique has been applied to many CPW-based artificial transmission lines and microwave devices based on them, as will be seen along this book). In CPWs, the transverse line geometry (see Fig. 1.2) is determined by the strip, W, and slot, G, widths (and of course by the substrate thickness, h, which is not usually a design parameter). Therefore, the characteristic impedance does not univocally determine the transverse line dimensions (W and G). This flexibility in the lateral geometry can be of interest in some applications. Moreover, for a given substrate thickness, it is possible to achieve higher characteristic impedances in CPW technology as compared to microstrip lines, where the strip width is univocally determined by the impedance value. Suspended microstrip lines can be implemented by using sustaining posts in order to create an air gap between the ground plane and the substrate, or by means of advanced micromaching technologies [18], where the (lossy) substrate is partly removed by etching. As compared to conventional microstrip lines, suspended microstrip lines are thus low-loss lines. Moreover, because most of the field is in the air gap, higher characteristic impedances can be realized. Additionally, the presence of the air gap reduces the effects of dispersion, and such lines are of special interest in the upper microwave and lower millimeter wave bands. However, despite these beneficial properties, suspended microstrip lines are difficult to implement, and their use is restricted to applications where the required performance justifies the higher fabrication costs, or to monolithic microwave integrated circuits (MMICs). A modification of the suspended microstrip line is the so-called inverted microstrip line, where the conductor strip is placed below the substrate, in contact with the air gap. The advantages and drawbacks are similar to those of the suspended microstrip line.

29

The slot mode of a CPW is, in general, an undesired mode of odd nature, that is, the symmetry plane of the CPW transmission line is an electric wall (or a virtual ground) for this mode. Conversely, the symmetry plane is a magnetic wall for the fundamental (even) CPW mode.

30

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

Slot lines are transmission lines that can be used either alone or in combination with microstrip lines on the opposite side of the substrate [19, 20]. Resonant slots coupled to microstrip lines have been used for the implementation of stop band filters, and slot antennas, consisting of a resonant slot fed by a microstrp line, are very well known [21]. For guided wave applications, radiation must be minimized. This is achieved through the use of high permittivity substrates, which causes the slot-mode wavelength to be small compared to free-space wavelength, and thereby results in the fields being closely confined to the slot with negligible radiation loss. The slot line shares with the CPW the coplanar configuration; hence, slot lines are especially convenient for shunt connecting lumped elements. Like microstrip lines or CPWs, slot lines do not support purely TEM modes. Indeed, the slot mode is markedly a non-TEM mode, and hence the characteristic impedance and the phase velocity in the slot line vary with frequency [19] (in contrast to microstrip lines or CPWs, where the line parameters are constant to a first-order approximation). Except the slot line, the planar transmission lines considered earlier are singleended (or unbalanced) lines. However, in applications where high immunity to noise, low crosstalk, and low electromagnetic interference (EMI) are key issues (i.e., in highspeed digital circuits), balanced (or differential) lines, and circuits are of primary interest. In two-conductor unbalanced transmission lines, the conductors have different impedance to ground, as sketched in Figure 1.11a. Such lines are fed by single-ended ports in which there is an active terminal and a ground terminal (i.e., one of the conductors is fed whereas the other is tied to ground potential). One of the conductors transports the signal current and the other acts as the return current path. By contrast, in two-conductor balanced lines (Fig. 1.11b), the conductors have equal potential with respect to ground and are in contra phase, and the currents flowing in the conductors have equal magnitude but opposite direction (each conductor provides the signal return path for the other). Such balanced lines are fed by a differential port, consisting of two terminals, neither of which being explicitly tied to ground. In balanced lines, the conductors have the same impedance to ground, this being the main relevant difference compared to unbalanced lines. Microstrip lines, CPWs, and striplines are examples of unbalanced lines. By contrast, slot lines, or coplanar strips (CPS), are balanced structures by nature. However, these balanced structures can be regarded as either balanced or unbalanced, depending on whether the excitation is balanced or unbalanced, respectively. Two-conductor balanced transmission lines can also be implemented by etching parallel strips at both sides of a dielectric slab

(a) + V1 –

(b) I1

I2

(c) I1

+ + V2 V1 I 1 – –

I2 I2

I1 + + V2 V1 I 1 – –

I2 I2

+ V2 –

Coupled-lines FIGURE 1.11 Schematic of two-port transmission lines. (a) Two-conductor unbalanced line, (b) two-conductor balanced line, and (c) three-conductor balanced line.

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

31

(see Fig. 1.2). This paired strips transmission line is useful, for instance, to feed antipodal printed dipole antennas [22]. Notice that the symmetry plane of the paired strips transmission line is an electric wall, and hence a virtual ground. Therefore, this structure can be analyzed by applying symmetry properties, that is, by removing the lower half of the structure, and adding a conducting plate, acting as ground plane, in the backside of the “sliced” substrate. Obviously, the resulting structure is a singleended microstrip transmission line, and hence the main line parameters are calculated by applying the same formulas. However, the voltage drop across the paired strips is twice the voltage drop in the microstrip transmission line, whilst the current is the same. This means that the characteristic impedance of the balanced paired strips line is twice the characteristic impedance of the microstrip line. Most practical implementations of balanced lines incorporate a ground plane, or some other global reference conductor. Such differential structures cannot be considered as pure two-conductor systems, since the ground plane becomes the third conductor of a three-conductor line (Fig. 1.11c). Such three-conductor line can be implemented by means of a pair of coupled lines over a ground plane. If the threeconductor line is not balanced due to the ground plane, currents flowing on it can unbalance the currents in the lines. On the contrary, if the three-conductor line is balanced, the active lines carry equal and opposite currents because the impedances of either line to ground are equal (see Fig. 1.11c). For instance, although a microstrip line is unbalanced, a two-port differential microstrip line can be designed by means of symmetric coupled lines (illustrated in Fig. 1.2—coupled microstrip) differentially driven. This balanced line consists of edge coupled lines that can be seen as a CPS line with a ground plane. 1.6 SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES There are several textbooks focused on the analysis and design of planar distributed circuits and antennas. Our aim in this Section is not to review all these transmission line applications, but to simply discuss some examples of distributed circuits where the involved transmission lines and stubs operate at different regimes, i.e., have different electrical lengths at the frequencies of interest. This includes planar circuits based on semi-lumped transmission lines (i.e., transmission lines with length l < λ/10), and based on λ/8, λ/4 and λ/2 lines. Some of the implementations reported in this Section will be later designed by means of artificial transmission lines in order to reduce circuit size, improve circuit performance, or achieve novel functionalities (or a combination of the previous beneficial aspects). 1.6.1 Semilumped Transmission Lines and Stubs and Their Application to Low-Pass and Notch Filters Let us start by considering the design of circuits on the basis of electrically small planar components, usually referred to as semilumped components. The characteristic

32

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

dimension30 of these components is typically smaller than λ/10 at the frequency of interest. The main relevant characteristic of semilumped components is the fact that they can be described by simple reactive elements, such as inductors, capacitors, or LC resonant tanks, up to frequencies satisfying the semilumped element approximation (l < λ/10). Let us consider an electrically small (l < λ/10) section of a transmission line, with electrical length βl and characteristic impedance Zo. From (1.6), we can write βl = ωl L C

1 91

The previous equation can be expressed in terms of the characteristic impedance and the per-unit-length inductance of the line as follows: βl = ωl L C = ωl

L Zo

1 92

or as a function of the characteristic impedance and the per-unit-length line capacitance according to βl = ωl L C = ωlZo C

1 93

Thus, the line inductance and capacitance of the considered transmission line section can be expressed as follows: Lω = Zo βl Cω =

βl Zo

1 94a 1 94b

Equations 1.94 reveal that if the line is electrically short and Zo is high, the capacitance of the considered transmission line section can be neglected and hence the line is essentially a series inductance; conversely, for an electrically short low impedance line, the line can be described by a shunt capacitance (the line inductance can be neglected). At first sight, one may erroneously deduce from (1.94) that the electrically short line condition (semilumped approximation) is not a requirement to describe the line by means of a series inductance or a shunt capacitance (very high/low value of Zo leads to a negligible line capacitance/inductance). However, the line is not free from distributed effects (despite the fact it has an extreme—high or low—characteristic impedance) if it is not electrically short. If the line is electrically short, expressions (1.33) can be approximated by

30

By characteristic dimension we mean the length (for a transmission line section or stub), the diameter (for a circular semilumped component, such as a circularly shaped split ring resonator or SRR), or the longest side length (for a rectangular planar component, such as a folded stepped impedance resonator or SIR). The analysis of these electrically small resonators and their applications will be considered later.

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

Zin ZL = ∞ = − j

Zo 1 = −j βl Cω

Zin ZL = 0 = jZo βl = jLω

33

1 95a 1 95b

where (1.94) has been used. Hence, it is demonstrated that an electrically small transmission line section with extreme characteristic impedance can be described either by a series inductor or by a shunt capacitor, and the element values can be inferred from (1.94).31 From the previous words, it follows that by cascading electrically small transmission line sections with high and low characteristic impedance, we can implement a ladder network with series inductances and shunt capacitances, that is, a low-pass filter. The design procedure simply consists of setting the high and low characteristic impedance to implementable values, and determining the line length l of each transmission line section by means of (1.94) [1]. Notice that the higher/lower the characteristic impedance, the shorter the resulting inductive/capacitive transmission line section. After calculation, it is necessary to verify that each transmission line section satisfies the semilumped element approximation. An illustrative example of a stepped impedance low-pass filter (in CPW technology) and its frequency response are presented in Figure 1.12. The device is a ninth-order Butterworth low-pass filter with a cut-off frequency of fc = 2 GHz (the element values of the filter are inferred from impedance and frequency transformation from the low-pass filter prototype [1]). The widths of the central strips and slots for the different sections are obtained by means of a transmission line calculator, once the characteristic impedance of the high- and low-impedance transmission line sections is set. Such calculators incorporate the formulas that link the lateral line geometry to the characteristic impedance, present in most textbooks focused on transmission lines [6]. In microwave engineering, shunt- and series-connected resonators (either distributed or semilumped) are key elements. In particular, shunt-connected series resonators introduce transmission zeros (notches) in the frequency response, which are of interest for harmonic suppression, or to improve the selectivity in microwave filters (elliptic low-pass filters are implemented by means of shunt resonators in order to generate transmission zeros above the pass band of interest). An open-ended shunt stub behaves as a parallel connected series resonator in the vicinity of the frequency that makes the line to be λ/4 long. However, the stub length can be significantly reduced by considering a stepped impedance topology, as depicted in Figure 1.13 [23]. Such element is known as stepped impedance shunt stub (SISS) and is described by a grounded series resonator. The narrow (high impedance) and wide (low impedance) transmission line sections correspond to the inductance and capacitance, respectively. The admittance of the SISS (seen from the host line) is given by

31

Alternatively, expressions (1.94) and the semilumped approximation requirement have been derived by considering the equivalent T-circuit model of a transmission line section, where the series and shunt impedance are calculated from the elements of the impedance or admittance matrix [1].

34

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

(a)

(b) 0

S21 (dB)

–10 –20 –30 –40 –50 –60 0

1

4 2 3 Frequency (GHz)

5

6

FIGURE 1.12 Order-9 Butterworth stepped impedance low-pass filter (a) and measured (solid line) and EM simulated (thin line) frequency response (b). The filter was fabricated on the Rogers RO3010 substrate with dielectric constant εr = 10.2, and thickness h = 1.27 mm. Filter length is 9.4 cm.

W1

l1

Z1, Φ1

Z2, Φ2

l2

W2 Z0 FIGURE 1.13 Topology of the SISS in microstrip technology and relevant dimensions (Z2 Z1).

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

YSISS = − j

tan ϕ1 + Ktan ϕ2 Z2 tan ϕ1 tan ϕ2 − Z1

35

1 96

where ϕ1 and ϕ2 are the electrical lengths of the low- and high-impedance line sections, respectively, and K=

Z1 Z2

1 97

is the impedance ratio of the SISS [24]. At resonance, the denominator in (1.96) vanishes, and the following condition results32: K = tanϕ1 tanϕ2

1 98

It is obvious from (1.98) that to minimize the total electrical length (ϕT = ϕ1 + ϕ2) of the resonator, K must be as small as possible (K 1) [24]. The reduction of ϕT is important for two reasons: (1) to reduce the length of the SISS and (2) to be able to describe the SISS by means of a lumped element model (grounded series LC resonator) over a wide frequency band. Under the assumption that the two transmission line sections of the SISS are electrically small, the tangents in (1.96) can be linearized, and the admittance of the SISS is found to be YSISS = − j

ϕ1 + Kϕ2 Z2 ϕ1 ϕ2 − Z1

1 99

This admittance is identical to that of an LC series resonant tank, given by YLC = − j

ωC LCω2 − 1

1 100

provided the following mapping is satisfied: C=

l1 l2 + = C1 + C2 vp1 Z1 vp2 Z2

l1 vp1 Z1 Z2 l2 C1 = L2 L= l2 vp2 l1 C1 + C2 + vp1 Z1 vp2 Z2

1 101

1 102

32 Notice that if K = 1, the resonance condition (1.98) rewrites as tan(ϕ1 + ϕ2) = ∞, giving ϕT = ϕ1 + ϕ2 = π/2, that is, a λ/4 open stub at resonance. This result is easily inferred by applying the following trigonometric identity to (1.96): tan(ϕ1 + ϕ2) = (tanϕ1 + tanϕ2)/(1 − tanϕ1 tanϕ2).

36

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

where vp1 and vp2 are the phase velocities of the low- and high-impedance transmission line sections, respectively, C1 and C2 are the line capacitances, namely, C1 =

l1 = C1 l1 vp1 Z1

1 103

C2 =

l2 = C2 l2 vp2 Z2

1 104

C i (i = 1, 2) being the per unit length capacitances of the lines, and L2 is the inductance of the high impedance transmission line section, that is, L2 = L2 l2

1 105

Notice that although C is dominated by the capacitance of the low impedance transmission line section, C1, the contribution of C2 on C may be nonnegligible if either K or l2 are not very small. It is also interesting to note that L is somehow affected by the capacitive line section, such inductance being smaller than the inductance of the highimpedance transmission line section, L2. Obviously, to a first-order approximation, the resonator elements are given by C ≈ C1 and L ≈ L2 (as expected on account of 1.95), although this approximation sacrifices accuracy. Figure 1.14 shows the photograph of a SISS resonator loading a 50 Ω microstrip transmission line, where L = 3.7 nH and C = 6.9 pF (the resonance frequency is fo = 1 GHz [23]). The impedance of the inductive line is that corresponding to a line width of 150 μm, namely, Z2 = 101.8 Ω (the structure is implemented on the Rogers RO3010 substrate with dielectric constant εr = 10.2 and thickness h = 1.27 mm). For the lowimpedance transmission line section, the width (23 mm) guarantees that the first transverse resonance occurs beyond 2fo (see details in Ref. [25]). This gives a characteristic impedance of Z1 = 5.8 Ω. The lengths of the lines, l1 = 2.8 mm and l2 = 4 mm, were derived from (1.101) and (1.102) (actually some optimization was required due to the effects of line discontinuities, not accounted for by the model). These line lengths correspond to electrical lengths of ϕ1 = 20.7 and ϕ2 = 23.8 at 2fo. The EM simulation and the measured frequency response of the SISS-loaded line is also depicted in Figure 1.14. The circuit simulation of the same line loaded with a shunt connected LC resonator with the reactive values given above is also included in the figure, for comparison purposes. The agreement is excellent up to 2fo, indicating that the semilumped approximation is valid in the considered frequency range. Although, typically, notch filters (i.e., stop band filters with a peaked response, or transmission zero) exhibit narrow stop bands, the structure of Figure 1.14 can be considered to belong to this category (the bandwidth can be controlled by the inductance/capacitance ratio). To end this subsection devoted to semilumped transmission lines and components, let us briefly consider the stepped impedance resonator (SIR) [26], which consists of a pair of wide transmission line sections sandwiching a narrow strip (Fig. 1.15a). This resonator is electrically smaller than the conventional λ/2 resonator (in the same form

37

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES (a)

(b)

(c) 0

200

S21 S11, S21 (degrees)

S11, S21 (dB)

–10 S11 –20

–30 –40 0.0

Series LC resonator SISS EM sim. SISS meas.

100

0

S21

–100

S11

–200 0.5

1.0 1.5 Frequency (GHz)

2.0

2.5

0.0

0.5

1.0 1.5 Frequency (GHz)

2.0

2.5

FIGURE 1.14 SISS-loaded microstrip line (a), insertion and return loss (b) and phase response (c). Reprinted with permission from Ref. [23]; copyright 2011 IET. (a)

FIGURE 1.15

(b)

Topology of a SIR (a), and folded SIR (b).

than the SISS is electrically smaller than the λ/4 open stub). SIRs are typically driven through electric coupling, though they can also be externally driven by means of a time varying magnetic field by simply folding the SIR topology, as shown in Figure 1.15b. The size of the folded SIR can be further reduced by decreasing the gap distance between the wide transmission line sections, since this introduces an extra capacitance to the structure. SIRs have been used in a wide variety of microwave applications, including applications involving artificial transmission lines (as will be later shown). To illustrate the potentiality of these resonators, an SIR-based order-3 elliptic-function low-pass filter is reported (see further details in Ref. [27]). The filter is based on a CPW transmission line loaded with an SIR etched in the back substrate side. The wide strip sections of the SIR are placed face-to-face with the central strip and ground plane of the CPW host line, resulting in a shunt connected series resonator which introduces a transmission zero. The circuit model of an elliptic low-pass filter

(a) L3

L1

L5

L2

L4 C4

C2

(b)

(c) b

CS G

CS L3

L1 L2

W

Ca

Cb

Ca C2 a

(d) 0 S11

S21 (dB), S11 (dB)

–10

–20 S21

–30

EM sim. Ideal circuit sim. Improved circuit sim.

–40

–50 1

2

3

4

5

6

7

Frequency (GHz) FIGURE 1.16 (a) Low-pass elliptic-function prototype filter with shunt connected series resonators (the circuit correspond to a fifth-order prototype), (b) topology of the SIR-based low-pass filter (order-3), (c) equivalent circuit model including parasitics, and (d) EM response, ideal filter prototype response and circuit response including parasitics. The considered substrate thickness and dielectric constant are h = 254 μm and εr = 11.2, respectively. Dimensions are W = 5 mm, G = 0.55 mm, a = 3.24 mm, b = 3.99 mm. Back side metal is indicated in black colour. The element values of the ideal prototype filter shown in (a) are L1 = L3 = 4.7 nH, L2 = 1.38 nH, C2 = 2.98 pF. The element values of the complete circuit model in reference to the circuit shown in (c) are L1 = L3 = 4.7 nH, L2 = 1.65 nH, C2 = 2.5 pF, Ca = 0.08 pF, Cb = 0.44 pF, Cs = 0.115 pF. With regard to parasitics, Cs models the capacitance associated to the meander, and Ca, Cb are the capacitances from the central strip to the ground plane. Reprinted with permission from Ref. [27]; copyright 2010 IEEE.

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

39

based on shunt connected series resonators,33 the proposed order-3 SIR-based filter, its circuit model including parasitics, and the frequency response, are depicted in Figure 1.16. The remarkable aspects of these SIR-based filters are the small size, and the excellent agreement between the ideal filter response (ideal circuit simulation) and the EM simulation up to frequencies above the transmission zero frequency. At higher frequencies, the parasitics must be included for an accurate description of the filter response (indicated as improved circuit simulation in the figure). Elliptic function low-pass filters using SISS in microstrip technology have also been reported [28]. 1.6.2 Low-Pass Filters Based on Richard’s Transformations Let us now consider the potential of λ/8 open and short-circuit stubs, which are electrically larger than the semilumped components considered in the previous section. According to Richard’s transformations, such stubs can be used to replace shunt inductors and capacitors. Let us illustrate their application to the implementation of low-pass filters. The key idea is to force the length of the stubs to be λ/8 at the filter cut-off frequency, ωc. Using (1.35), the reactance of the short-circuit and open-circuit stub at ωc is forced to be identical to that of the inductor and capacitor, respectively. Below that frequency, we also expect a similar reactance because the stubs have roughly a linear dependence with frequency. However, discrepancies between the reactances of the stubs and lumped elements are expected above ωc.34 Let us consider the implementation of an order-3 Chebyshev low-pass filter with a cut-off frequency of fc = 2 GHz, and 0.5 dB ripple. From impedance and frequency transformation from the low-pass filter prototype, the series inductances and the shunt capacitance of the filter are L1 = L3 = 6.35 nH, and C2 = 1.74 pF, respectively. Application of (1.35) gives Zo1,3 = 79.8 Ω and Zo2 = 45.6 Ω, for the inductive and capacitive stubs, respectively. Since the implementation of series stubs is complex, at least in microstrip technology, let us replace the inductive stubs with shunt stubs. To this end, the Kuroda identity shown in Figure 1.17 is used [1].35 To apply the Kuroda identity of Figure 1.17, we first cascade a pair of λ/8 long 50 Ω transmission line sections at the input and output port of the filter. Notice that this has effects on the phase response of the filter, but not on the magnitude of the insertion and return loss. Application of the Kuroda identity leads to the circuit of Figure 1.18. The layout of this filter, for microstrip technology considering the Rogers R04003C substrate with dielectric constant εr = 3.38 and thickness h = 0.81 mm, is also depicted in the Figure. The EM simulation of the filter response is compared with the ideal Chebyshev filter response, where it can be appreciated that the agreement is excellent up to the 33

Alternatively, elliptic-function lowpass filters can be implemented by cascading series connected parallel resonators and shunt capacitors. 34 Since the reactance of the stubs is a periodic function with frequency, the frequency response of the filter is also a periodic function, and spurious (or harmonic) bands are generated. 35 Kuroda’s identities are equivalences between two-port networks containing series- or shunt-reactive elements and transmission line sections (called unit elements), that are used to physically separate transmission line stubs, to transform series stubs into shunt stubs, and to change unrealizable characteristic impedances into implementable ones.

40

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

Zo1

l = λ/8

l

n2 = 1+

Zo2

n2Zo2

Zo1 n2Zo1

Zo2 Port 1

l

Port 1

Port 2

l

l Port 2

FIGURE 1.17 Kuroda identity used for the design of the filter of Figure 1.18.

cut-off frequency, and the frequency selectivity of the distributed implementation is significantly better, although with the presence of spurious bands (the filter response repeats every 8 GHz), as predicted before. 1.6.3 Power Splitters Based on λ/4 Lines Power splitters, combiners, and couplers are fundamental building blocks in RF/ microwave engineering. Most of their simplest implementations as distributed circuits are based on λ/4 lines.36 Let us, hence, illustrate the application of λ/4 lines to the implementation of power splitters (it will be later shown that these lines can be replaced with artificial lines in order to reduce splitter size and obtain dual-band functionality). Power splitters are reciprocal37 three-port networks (or multiport networks if the number of output ports is higher than 2) with a matched input port, that is, there is not power return to this port if the output ports are terminated with matched loads. If the splitter is lossless, the scattering matrix (for the case of a 1:2 device) can be written in the general form [1]:

S=

36

0 α

α

α γ

−γ

α −γ

γ

1 106

Power splitters can also be implemented by means of lumped resistive elements [1]. Such splitters ideally exhibit an infinite operational bandwidth, but they are lossy. By contrast, distributed splitters can be considered (to a first approximation) lossless, but their functionality is restricted to a certain band in the vicinity of the operational frequency. 37 Reciprocal networks are defined as those networks verifying that the effects of a source, located at one port, over a load, located at another port, are the same if the source and load interchange the ports where they are connected [1]. In reciprocal networks, the scattering matrix is symmetric.

41

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

l = λ/8

l = λ/8

(a) Port 1

130Ω

50Ω

Port 2

81.3Ω

45.6Ω

81.3Ω

50Ω

130Ω

l = λ/8

l = λ/8

l = λ/8

(b) 1.77

12.15

0.17

11.2

11.7 0.7

(c)

2.1

0

│S11│, │S21│ (dB)

S21

–20 S11 –40 Electromagnetic simulation Circuit simulation Chebyshev response

–60 0

2

4 Frequency (GHz)

6

8

FIGURE 1.18 Schematic (a), layout (b), and frequency response (c) of the low-pass filter based on Richard’s transformations. The relevant dimensions (in mm) are indicated. The circuit simulation in (c) was obtained by using a commercial circuit and schematic solver, where the transmission lines and stubs are modeled by the corresponding distributed models.

with α = 1 2 and γ = 1 2. The two canonical forms of lossless symmetric power splitters are depicted in Figure 1.19, where the impedances of the inverters (λ/4 lines), indicated in the figure, are derived by forcing the matching condition for the input port. In both implementations of Figure 1.19, α = − j 2, whereas γ = 1/2 for Figure 1.19a and γ = −1/2 for Figure 1.19b. The number of output ports of the splitter

42

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

(a)

(2) ϕ = π/2

(b)

Z0

√2Z0 (1)

(1)

Z0

Z0

Z0 /√2

ϕ = π/2

(2) Z0 Z0 (3)

√2Z0 ϕ = π/2

Z0 (3)

FIGURE 1.19 Canonical forms of the two-output distributed symmetric power splitter. (a) With two inverters and (b) with one inverter.

can be arbitrarily large. In order to preserve matching, the characteristic impedance of the inverters must be Zo = 50 nΩ for the structure of Figure 1.19a and 50 nΩ for the structure of Figure 1.19b, where n is the number of output ports. A 1:3 power divider, corresponding to the configuration of Figure 1.19a, and its EM response are shown in Figure 1.20. The device was designed to be operative at 1.5 GHz, as revealed by the good matching at that frequency. 1.6.4 Capacitively Coupled λ/2 Resonator Bandpass Filters The last illustrative example is a bandpass filter based on λ/2 transmission lines acting as distributed resonators. If a gap is etched on a transmission line, the frequency response is a high-pass type response, and this can be described by series connecting a capacitor to the line. However, if two capacitive gaps are etched in the line, rather than a high-pass response with enhanced rejection in the stop band, the structure exhibits a bandpass response that can be attributed to a resonance phenomenon. At the frequency where the distance between gaps is roughly λ/2, the forward and backward travelling waves (caused by gap reflections) in the resonator sum up in phase, and small coupling (and hence power transfer) between the feeding line and the resonator is enough to achieve total power transmission (assuming lossless lines) between the input and the output ports. Of course, this resonance phenomenon occurs at frequencies satisfying l = nλ/2 (with n = 1, 2, 3, …).38 Based on this phenomenon, bandpass filters with controllable response can be implemented (see Ref. [1] for further details on the design of this type of filters). As an example, an order-3 bandpass filter in 38

Actually, this is the resonance condition for an unloaded resonator. The gaps introduce some phase shift in the reflected waves, and the resonance condition is slightly modified.

43

SOME ILLUSTRATIVE APPLICATIONS OF PLANAR TRANSMISSION LINES

(a)

(2)

0.2

1.04

19.7

(1)

(3)

(4) (b) │S11│, │S21│, │S31│, │S41│ (dB)

0

S21, S31, S41

–10

S11

–20

–30 Electromagnetic simulation Circuit simulation

–40 0

1

2

3

Frequency (GHz) FIGURE 1.20 Example (layout) of a power splitter (a), and frequency response (b). Relevant dimensions (in mm) and device ports are indicated. The width of the three λ/4 lines gives a characteristic impedance of Zo = 50 3 = 86 6 Ω. The considered substrate is the Rogers RO3010 with dielectric constant εr = 10.2 and thickness h = 1.27 mm.

44

FUNDAMENTALS OF PLANAR TRANSMISSION LINES

(a) 1.09

8.14 0.06

7.89

0.75

(b)

│S11│, │S21│ (dB)

0

S11

–20

–40 S21 –60 0

Capacitively-coupled TLs Lossless EM simulation Lossy EM simulation 5 10 Frequency (GHz)

15

FIGURE 1.21 Example of a capacitively coupled λ/2 resonator bandpass filter (a) and frequency response (b). Relevant dimensions (in mm) are indicated. The considered substrate is the Rogers RO3010 with dielectric constant εr = 10.2 and thickness h = 1.27 mm.

microstrip technology and its frequency response are shown in Figure 1.21. Resonator lengths and inter-resonators distance (i.e., gap space) have been calculated in order to obtain a Chebyshev response with 0.5 dB ripple, central frequency fo = 6 GHz, and 5% fractional bandwidth (it will be shown in Chapter 2 that resonator’s length can be reduced by means of slow wave artificial transmission lines). REFERENCES 1. 2. 3. 4.

D. M. Pozar, Microwave Engineering, Addison Wesley, Reading, MA, 1990. B. C. Wadell, Transmission Line Design Handbook, Artech House, Norwood, MA, 1991. J. C. Freeman, Fundamentals of Microwave Transmission Lines, John Wiley, New York, 1996. F. Di Paolo, Networks and Devices Using Planar Transmission Lines, CRC Press, Boca Raton, FL, 2000. 5. P. C. Magnusson, A. Weisshaar, V. K. Tripathi, G. C. Alexander, Transmission Lines and Wave Propagation, CRC Press, Boca Raton, FL, 2001. 6. I. Bahl and P. Barthia, Microwave Solid State Circuit Design, 2nd Edition, John Wiley, New York, 2003.

REFERENCES

45

7. R. K. Mongia, I. J. Bahl, P. Barthia, and J. Hong, RF and Microwave Coupled-Line Circuits, 2nd Edition, Artech House, Norwood, MA, 2007. 8. L. Ganesan, and S. S. Sreeja Mole, Transmission Lines and Waveguides, 2nd Edition, McGraw Hill, New Delhi, 2010. 9. K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. Microw. Theory Technol., vol. MTT-13, pp. 194–202, 1965. 10. P. I. Richards, “Resistor-transmission-line circuits,” Proc. IRE, vol 36. pp. 217–220, 1948. 11. S. Ramo, J. R. Whinnery, and T. Van Duzer, Field and Waves in Communication Electronics, 3rd Edition, John Wiley, New York, 1994. 12. R. A. Pucel, D. J. Massé, and C. P. Hartwig, “Losses in microstrip,” IEEE Trans. Microw. Theory Technol., vol. MTT-16, pp. 342–350, 1968. 13. G. Zou, H. Gronqvist, P. Starski, and J. Liu, “Characterization of liquid crystal polymer for high frequency system-in-a-package applications,” IEEE Trans. Adv. Packag., vol. 25, pp. 503–508, 2002. 14. K. Chang, Microwave Ring Circuits and Antennas, John Wiley, New York, 1994. 15. L. Yang, A. Rida, and R. Vyas, M. M. Tentzeris “RFID tag and RF structures on a paper substrate using inkjet-printing technology,” IEEE Trans. Microw. Theory Technol., vol. 55, pp. 2894–2901, 2007. 16. Association Connecting Electronics Industries. IPC TM-650 2.5.5.13. Relative Permittivity and Loss Tangent Using a Split-Cylinder Resonator. http://www.ipc.org/TM/2-5-5-13.pdf. Accessed February 5, 2015. 17. M. D. Janezic and J. Baker-Jarvis, “Full-wave analysis of a split-cylinder resonator for nondestructive permittivity measurements,” IEEE Trans. Microw. Theory Technol., vol. 47, pp. 2014–2020, 1999. 18. C. T.-C. Nguyen, L. P. B. Katehi, and G. M. Rebeiz, “Micromachined devices for wireless communications,” Proc. IEEE, vol. 86, pp. 1756–1768, 1998. 19. S. B. Cohn, “Slot line on a dielectric substrate,” IEEE Trans. Microw. Theory Technol., vol. MTT-17, pp. 768–778, 1969. 20. K. C. Gupta, R. Carg, I. Bahl, and P. Barthia, Microstrip Lines and Slotlines, 2nd Edition, Artech House, Norwood, MA, 1996. 21. R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook, Artech House Inc., Norwood, 2001. 22. F. J. Herraiz-Martínez, F. Paredes, G. Zamora, F. Martín, and J. Bonache, “Dual-band printed dipole antenna loaded with open complementary split-ring resonators (OCSRRs) for wireless applications,” Microw. Opt. Technol. Lett., vol. 54, pp. 1014–1017, 2012. 23. J. Naqui, M. Durán-Sindreu, J. Bonache, and F. Martín, “Implementation of shunt connected series resonators through stepped-impedance shunt stubs: analysis and limitations,” IET Microw. Antennas Propag., vol. 5, pp. 1336–1342, 2011. 24. M. Makimoto and S. Yamashita, “Compact bandpass filters using stepped impedance resonators,” Proc. IEEE, vol. 67, pp. 16–19, 1979. 25. T. C. Edwards and M. B. Steer, Foundations of Interconnect and Microstrip Design, 3rd Edition, John Wiley, New York, 2000. 26. M. Makimoto and S. Yamashita, “Bandpass filters using parallel-coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Technol., vol. MTT-28, pp. 1413– 1417, 1980.

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FUNDAMENTALS OF PLANAR TRANSMISSION LINES

27. M. Durán-Sindreu, J. Bonache, and F. Martín, “Compact elliptic-function coplanar waveguide low-pass filters using backside metallic patterns,” IEEE Microw. Wireless Compon. Lett., vol. 20, pp. 601–603, 2010. 28. G. Matthaei, L. Young, and E.M.T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, Artech House, Norwood, MA, 1980.

2 ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

2.1

INTRODUCTION AND SCOPE

In the framework of RF/microwave engineering, one-dimensional periodic structures are transmission lines and waveguides periodically loaded with identical elements (lumped, semilumped, inclusions, defects, etc.), or with a periodic perturbation in their cross-sectional geometry (nonuniform transmission lines and waveguides).1 The main relevant properties of one-dimensional periodic structures, which will be reviewed in this chapter, can be useful for the implementation of artificial transmission lines with various functionalities and applications based on them. Specifically, periodic transmission lines exhibit stop/pass bands, and support the propagation of waves with phase velocities lower (slow waves) or higher (fast waves) than the speed of light. Thus, transmission lines based on periodic structures can be applied to the implementation of filters, reflectors, electromagnetic bandgaps (EBGs),2 slow wave structures

1 Two-dimensional and three-dimensional periodic structures can also be of interest at RF/microwave (and even at optical) frequencies, in applications such as frequency-selective surfaces, antenna substrates and superstrates (to improve antenna performance), isolators, and so on, but these structures are out of the scope of this book, which is focused on artificial transmission lines. 2 In analogy with semiconductor crystals (which exhibit forbidden energy bands, or gaps), periodic structures in the optical domain are usually identified as photonic crystals (PCs), or photonic bandgaps (PBGs). This explains the term “electromagnetic bandgap” (EBG) used to designate such structures at RF/microwave frequencies.

Artificial Transmission Lines for RF and Microwave Applications, First Edition. Ferran Martín. © 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc.

48

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

(of interest for device miniaturization), and leaky wave antennas (LWAs), among others. The purpose of this chapter is to briefly present the Floquet (or Bloch mode) analysis of one-dimensional periodic structures (which will bring us to the concept of space harmonics), and the transfer matrix method, applied to the unit cell, for obtaining the modal solutions (or dispersion curves) of the fundamental space harmonic. This will be important to predict the frequency response of the considered lines based on periodic structures. Moreover, an alternative and complementary analysis for periodic transmission lines with nonuniform cross section, (e.g., microstrip lines with defected ground planes or strip width modulation), based on the coupled mode theory, is also included in this chapter. By means of justified approximations, it will be shown that this analysis provides valuable information (through analytical expressions) relative to the main relevant parameters of these lines (S-parameters, maximum reflectivity and bandwidth of the stop bands, etc.). The periodic transmission lines that will be considered and studied in this chapter through the earlier-cited approaches (the transfer matrix method and the coupled mode analysis) include EBG-based transmission lines (either with defected ground planes or strip width modulation), and transmission lines loaded with reactive elements. The main applications of these lines will also be reviewed in the chapter.

2.2

FLOQUET ANALYSIS OF PERIODIC STRUCTURES3

Let us consider a one-dimensional infinite periodic structure (transmission line or waveguide) with period l, and the propagation axis denoted as z. If the time dependence is chosen as e jωt, and if the cross-sectional dependence is suppressed, the Floquet’s theorem states that the fields propagating along the line can be expressed as Bloch waves according to [1–5]: Ψ z = e −γz P z

21

where γ is the propagation constant, and P(z) is a periodic function with period l: P z+l =P z

22

Thus, the field behavior can be expressed in terms of a fundamental traveling wave, with propagation constant γ, and a standing wave P(z), which repeats in each unit cell

3 This section is mainly based on a short course given by Arthur A. Oliner, who recently passed away, and who was a pioneer in the topic of periodic structures and leaky waves. His contributions on this topic are so well-written and comprehensible that can be effortlessly understood. Let us consider this section, extracted from this short course (published in Ref. [2]), as a tribute to him.

49

FLOQUET ANALYSIS OF PERIODIC STRUCTURES

and represents the local variations due to the periodicity. From (2.1), it follows that the fields at positions separated by one period are related by: Ψ z + l = e − γ z + l P z + l = e −γl Ψ z

23

or, in other words, the fields of a Bloch wave repeat at each unit cell terminal, having a propagation factor e−γl. Since P(z) is a periodic function, it can be expanded in a Fourier series as n = +∞

2πn l z

Pn e − j

Pz =

24

n = −∞

By inserting (2.4) into (2.1), the fields can be expressed as a superposition of traveling waves of the form n = +∞

Ψ z =

Pn e − j

β + 2πn l z

e − αz

25

n = −∞

where the propagation constant has been decomposed into the phase and attenuation constants, γ = α + jβ. The components of (2.5) are called space harmonics, in analogy to the harmonic decomposition of a periodic signal in time domain [2, 4]. The phase constants of the space harmonics are thus given by: βn = β +

2πn , n = 0, ± 1, ± 2, … l

26

and the phase of the fundamental harmonic (n = 0) is simply β. It is important to mention that the space harmonics do not exist independently. Rather than being modal solutions by themselves, they represent individual contributions to the whole field. Since the phase constants of the space harmonics differ by a constant, it follows that the group velocity is the same for all harmonics, that is, dω dβ

27

ω β + 2πn l

28

vgn = whereas the phase velocities are given by vpn =

From the space harmonics representation of the periodic structure, it is possible to infer the pass/stop band characteristics inherent to periodicity. The reason is mode interference (or coupling) between modes with similar phase velocities but opposite

50

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

group velocities (this aspect will be studied in Section 2.4 in detail). This results in stop bands (or gaps) in the frequency response. To gain insight on this effect, let us consider that a transmission line is slightly (periodically) perturbed so that the phase constant β of the fundamental harmonic can be approximated by a straight line (see expression 1.6). This means that the β − ω diagrams (or dispersion curves) of the set of space harmonics, obtained by displacing the dispersion diagram of the fundamental harmonic a quantity 2πn/l, are also straight lines, as depicted in Figure 2.1. This diagram points out that there are points where the straight lines cross, giving rise to mode coupling and hence stop bands, as Figure 2.2 illustrates. Thus, Figure 2.2 depicts the actual dispersion curves of the slightly perturbed transmission line. The gap bandwidths and the specific frequency dependence of the phase constants

ω

−4π

−2π

0

2π

4π

βl

FIGURE 2.1 β-ω diagram of the fundamental harmonic (solid line) and high-order space harmonics (dashed lines) for a transmission line with an infinitesimal periodic perturbation. ω

−4π

−2π

0

2π

4π

βl

FIGURE 2.2 Dispersion diagram of a one-dimensional periodic structure with space harmonics. The limits of the first Brillouin zone are indicated by vertical dotted lines. The region where leaky wave radiation is possible is delimited by the dashed lines with slopes − c/l and + c/l (and indicated by the gray-white alternating strips).

FLOQUET ANALYSIS OF PERIODIC STRUCTURES

51

(identical for all the space harmonics) are related to the nature and magnitude of the perturbation or loading element. Nevertheless, in view of Figure 2.2, the band gaps appear at frequencies satisfying βl = π (or l = λ/2), that is, f=

vp 2l

29

and its harmonics, vp being the phase velocity of the unperturbed line. Expression (2.9) is known as the Bragg condition, and states that the first stop band caused by the effects of periodicity4 appears at the frequency satisfying that the period is half the wavelength of the unperturbed line, λ. The presence of additional band gaps depends on the harmonic content of the Fourier expansion (2.4), which in turn depends on the specificities of the loading elements or perturbation (this aspect will also be considered in Section 2.4). It is remarkable that the part of the β − ω diagram of Figure 2.2 comprised between − π < βl < π is repetitive every βl = 2π along the abscissa axis. Usually, this region is referred to as the first Brillouin zone, and the curves within this region give full information of the dispersion characteristics of the periodic structure [1, 5]. It is also important to mention that if we consider either an infinitely long or terminated (with a matched load) periodic transmission line (so that reflections are avoided) fed by a source at one end, the waves excited by the source must carry their energy (and hence the group velocity) in the direction against the source (i.e., toward the load if it is present). Thus, in such situation, only the portions of the curves of Figure 2.2 with positive slope (or group velocity) are of interest.5 Regardless of the propagation direction, there are regions of the dispersion curves where β(ω) dβ(ω)/dω < 0. In these regions, the phase and group velocities are of opposite sign, and the corresponding waves are called backward waves.6 Another relevant aspect of one-dimensional periodic structures is the fact that these structures may radiate if they are open. The reason is that there are portions of the dispersion curves where the phase velocity is higher than the speed of light in vacuum, c, (fast wave regions, see Fig. 2.2). These radiating harmonics are leaky waves, and exhibit properties similar to leaky waves in uniform structures. A detailed analysis of leaky wave radiation in periodic structures is out of the scope of this book (see [6–9] for further details).7 Therefore, we will simply present a brief and straightforward 4 Stop bands not related to periodicity may appear at lower frequencies, for instance, by periodically loading a transmission line with resonators coupled to it. In this case, the first stop band is centered at the resonance frequency of the resonant element. 5 It is assumed that the energy travels toward the positive z-direction. 6 The concept of backward wave will be considered in detail in Chapter 3. Nevertheless, it is important to distinguish them from backward travelling waves. In backward waves, the energy propagates in the opposite direction to the phase of the waves, and it is therefore an unconventional type of propagation. Thus, for energy transmission in the positive z-direction, the phase of the waves propagates in the negative/positive z-direction for backward/forward waves. Backward/forward travelling waves are simply forward waves propagating in the negative/positive z-direction. 7 Nevertheless, we dedicate a subsection in Chapter 4 devoted to LWAs based on metamaterial transmission lines.

52

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

(a)

(b) x

ktn

x

k θ 0 kn

k ktn θ kn 0

z

z

FIGURE 2.3 Diagrams illustrating the generation of a forward (a) and backward (b) leaky wave in a periodic structure.

analysis to understand the phenomenon. Let us consider that the fields in the open periodic structure vary in both the longitudinal and transverse directions. The freespace wavenumber k is related to the parameters of each space harmonic as [2]8: 2 = β+ k2 = kn2 + ktn

2πn − jα l

2

2 + ktn

2 10

where it has been assumed that kn = βn − jα (component in the longitudinal direction) and ktn = βtn − jαtn (component in the transverse direction). To graphically illustrate the generation of leaky waves by the fast wave space harmonics, let us assume that the longitudinal attenuation constant α is negligible (i.e., kn = βn). In that case, if kn2 > k2 (slow wave regions), ktn is imaginary, and the mode does not leak. Conversely, if kn2 < k2 (fast waves), ktn is real and the mode radiates. This situation is illustrated in Figure 2.3, where it can be seen that for 0 < kn < k and − k < kn < 0, the radiation is forward and backward, respectively. For the specific frequencies where kn = 0, kn = k and kn = − k , leaky wave radiation is designated as broadside, endfire and backfire, respectively. Indeed, the direction of the radiated beam can be easily inferred from: cosθ =

kn k

2 11

which results from direct inspection to Figure 2.3. Let us now take into account the attenuation constants in (2.10). Through separation of the real and imaginary parts, we obtain: 2 k 2 = kn2 + ktn = β2n − α2 + β2tn − α2tn − 2j αβn + αtn βtn

2 12

8 The free-space wavenumber is denoted as k, rather than k0 (as usual), to avoid confusion with the fundamental space harmonic. However, in Section 2.4, we recover the usual designation (see expression 2.45) since in that section k is used as a parameter related to the weighting factors of the coupling coefficient (to be defined later), according to expression (2.66).

53

THE TRANSFER MATRIX METHOD

Since the imaginary part of the free-space wavenumber k is zero, it follows that αtn = −α

βn βtn

2 13

Expression (2.13) is interesting because it points out a significant difference between the forward and backward leaky waves. Namely, since the longitudinal attenuation constant α and transverse phase constant βtn are both positive, the transverse attenuation constant αtn is of opposite sign to the longitudinal phase constant of the space harmonic βn. Thus, if βn < 0, αtn is positive, meaning that the radiated fields decay in the transverse direction. Conversely, if βn > 0, αtn is negative, and the fields increase in the transverse direction. In this latter case (forward wave radiation), since this wave type would diverge at infinity in the transverse plane if it were defined everywhere, it cannot be spectral, that is, a proper mode. Therefore forward leaky waves are said to be non-spectral, contrary to backward leaky waves, which are called spectral [2]. By introducing (2.13) into (2.12), the relation between k, βn and βtn is found to be: β2n =

k 2 − β2tn + α2 α2 1− 2 βtn

2 14

Notice that although forward (nonspectral) leaky waves are mathematically improper,9 they have physical meaning and are useful for the implementation of radiating elements (LWAs). In practice, the leaky wave is defined only within a wedge-shaped region determined by the position of the source, where the leaky wave decays at all angles from the source. This solves the above inconsistency between the mathematical solution and the “physical” leaky waves, never exhibiting progressively increasing fields in the transverse direction.

2.3

THE TRANSFER MATRIX METHOD

Although the space harmonics are fundamental to explain several properties of periodic structures (the weight of each harmonic depends on the nature and magnitude of the perturbation), usually the fundamental space harmonic is dominant, and it suffices for the description of many structures (especially in the pass band regions, far enough from the band gap edges). In order to obtain the modal solutions (or dispersion characteristics) of the fundamental space harmonic (or equivalently of the Bloch wave), the transfer matrix method is very useful in situations where the fields at two positions separated by a period (related by Eq. 2.3) can be expressed as mutually dependent through a certain transfer function (or matrix), characteristic of the unit cell structure. 9 The increasing field in the transverse direction violates the boundary condition at infinity since the mathematical description of leaky waves holds throughout all space.

54

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

In

In–1 + Vn–1 −

A

B

C

D

In+1

+ Vn −

A

B

C

D

+ Vn+1 −

A

B

C

D

FIGURE 2.4 Periodic structure with unit cell described by the transfer ABCD matrix.

2.3.1 Dispersion Relation Let us consider the periodic structure depicted in Figure 2.4, where the unit cells are represented by boxes, and let us consider that the voltages and currents at the reference planes between adjacent unit cells are well (uniquely) defined10 and measurable quantities. The voltages and currents on either side of the nth unit cell are related by the ABCD matrix (see Appendix C) according to Vn In

=

A B

Vn + 1

C D

In + 1

2 15

On the other hand, according to the Floquet’s theorem, the voltages and currents at the n and n + 1 planes only differ by the propagation factor, that is Vn + 1 = e − γl Vn

2 16a

In + 1 = e − γl In

2 16b

From (2.15) and (2.16), it follows that Vn In

10

=

A B

Vn + 1

C D

In + 1

=

e γl Vn + 1 e γl In + 1

2 17

Strictly speaking, this uniqueness of voltages and currents is only possible for TEM modes, but it can also be made extensive to quasi-TEM modes. For non-TEM modes, the voltage and the current are not properly (uniquely) defined. However, it is possible to define an equivalent voltage and current that make these variables (and even their ratio, the impedance) useful quantities. To this end, the following considerations are applied [10]: (i) voltage and current are only defined for a particular mode, and are defined so that the voltage and current are proportional to the transverse electric and magnetic fields, respectively; (ii) the equivalent voltages and currents should be defined so that their product gives the power flow of the mode; and (iii) the voltage to current ratio for a single travelling wave should be equal to the characteristic impedance of the mode. Nevertheless, unless otherwise specified, the periodic structures considered throughout this book can be described, to a first-order approximation, by considering only the TEM or quasi-TEM modes. Therefore, expression (2.15) involves well-defined variables in our case.

55

THE TRANSFER MATRIX METHOD

or A− e γl

B

Vn + 1

C

D −e γl

In + 1

=0

2 18

Notice that, according to (2.17) and (2.18), the voltages and currents propagating in the line are the eigenvectors, whereas the propagation factor is given by the eigenvalues, or eigenmodes, of the system. For a nontrivial solution, the determinant of the matrix in (2.18) must be zero, namely AD + e2γl − A + D e γl − BC = 0

2 19

Since for a reciprocal system AD − BC = 1 (see Appendix C), (2.19) can be expressed as follows11: e γl + e − γl = A + D

2 20

and the dispersion relation can be finally written as cosh γl =

A+D 2

2 21

In a lossless and reciprocal periodic structure, the right-hand side of (2.21) is purely real. This means that the propagation constant is either purely real (γ = α, β = 0) or purely imaginary (γ = jβ, α = 0).12 In the first case, the Bloch wave is attenuated along the line, and the corresponding regions define the stop bands of the structure. If γ = jβ and α = 0, cosh(γl) = cos(βl), and (2.21) rewrites as follows: cos βl =

A+D 2

2 22

Expression (2.22) is thus valid in the propagation regions, where the modulus of the right-hand side is smaller than 1. If the unit cell of the periodic structure is symmetric with respect to the plane equidistant from the input and output ports, A = D and (2.22) can be simplified to cos βl = A

11

2 23

Reciprocity is assumed throughout this chapter. As it will be shown in Chapter 3, under some circumstances, lossless periodic structures may support modes that appear as conjugate pairs, that is, modes of the form γ = α ± jβ. Such modes are called complex modes. 12

56

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

2.3.2 Bloch Impedance Another important parameter is the relationship between the voltage and current at any position (plane) of the periodic structure. Such parameter can be inferred from (2.18), namely, A −e γl Vn + 1 + BIn + 1 = 0

2 24

Vn + 1 B , n =− A − e γl In + 1

2 25

and it follows that

Expression (2.25) does not depend on the plane where such voltage to current relation is calculated. It resembles the characteristic impedance of a transmission line, defined as the relation between voltage and current for a single propagating wave at any position in the line. However, since the propagating waves in the periodic structure are Bloch waves, it is more convenient to identify the impedance given by (2.25) as the Bloch impedance, ZB. Isolating eγl from (2.20) and introducing it into (2.25), it follows that the Bloch impedance has two solutions: ZB± = −

2B A−D

A + D 2 −4

2 26

one corresponding to forward traveling waves and the other to backward traveling waves. In general, the two solutions of (2.26) are complex. In the propagation regions, (A + D)2 < 4, and the resulting solutions have the same magnitude, identical imaginary part and real parts of opposite sign. In the forbidden (band gap) regions, the two solutions are purely imaginary13 and exhibit different magnitude, unless the unit cell is symmetric. In this case, A = D and (2.26) is simplified to ZB± = ±

B A2 − 1

2 27

For a lossless and symmetric structure, the two solutions of the Bloch impedance in the allowed regions are real and have opposite signs. Such different signs are indicative of propagation in the forward or backward direction (the negative sign for backward traveling waves is related to the definition of the currents in Fig. 2.4). Let us discuss the meaning of the complex Bloch impedance that results in the allowed bands of lossless periodic structures with asymmetric unit cells. According 13 In the forbidden regions, (A + D)2 > 4, hence the denominator in (2.26) is a real number. Since for a lossless structure B is purely imaginary, it follows that the Bloch impedance is purely imaginary in those regions.

57

THE TRANSFER MATRIX METHOD

to expression (1.8), in a lossless transmission line, the characteristic impedance is purely real. However, the Bloch impedance in lossless periodic structures is complex, if the unit cell is asymmetric. This complex impedance cannot be related to attenuation losses if the periodic line is lossless. The origin of the complex Bloch impedance in lossless periodic structures with asymmetric unit cells is intimately related to the termination of the line. If the unit cell is asymmetric, a finite periodic structure consisting of a certain number of cascaded unit cells is also asymmetric. Therefore, the input impedance of such an asymmetric structure terminated by certain load impedance depends on the locations (left or right) of the source and the load. In particular, if the structure is terminated with the Bloch impedance, the impedance seen from the input port is also the Bloch impedance. However, since the input impedance depends on the position of the source (input port), it follows that the Bloch impedance must be different for forward and backward traveling waves in asymmetric periodic transmission lines. Nevertheless, for an infinite periodic structure, where the effects of terminations vanish, propagation in the forward or backward directions must be undistinguishable. Indeed, if the structure is infinite, it can be described by a cascade of symmetric unit cells, by simply shifting the reference planes, as described in Figure 2.5.14 Since the Bloch impedance for this symmetric unit-cell-based structure in the propagation regions is real, we can conclude that the complex Bloch impedance in the asymmetric structure is caused by a phase shift in only one of the variables (voltage or current), as compared to the symmetric case (where voltage and current are in phase), as consequence of a displacement of the reference planes (this situation is also illustrated in Fig. 2.5). From this analysis, we can also conclude that for both forward and backward traveling waves propagating in the periodic structure with asymmetric unit cells, the real part of the Bloch impedance must be identical, whereas the imaginary parts must have identical magnitude and different sign (for the validity of this statement, we have assumed that the backward traveling waves exhibit positive current in the backward direction). To gain insight on the effects of asymmetry, let us consider a periodic structure that can be described by the network of Figure 2.5, where the dashed lines are the reference planes for the symmetric unit cell described by a T-circuit model, whereas the dotted lines are the reference planes of the asymmetric unit cell described by an L-circuit.15 The elements of the ABCD matrix for the unit cells considered in Figure 2.5 can be easily inferred [10]. In particular, for the symmetric unit cell: A=D=1+

14

Zs 2Zp

2 28a

This statement is based on the fact that any two-port network can be described by means of an equivalent π- or T-circuit. Despite the asymmetry of the equivalent π- or T-circuit of any asymmetric unit cell, by cascading such circuits, it is possible to describe any infinite structure by means of symmetric unit cells by simply shifting the reference planes. However, in a physical periodic structure, it is not necessarily possible to identify a symmetric unit cell. 15 An L-circuit can be considered a particular case of a T-circuit where one of the series impedances is null.

58

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

Zs

Zs

Zs

Zp

Zp

nL–1 nT–1

nL

nT

InL

InT

Zs/2

Zs/2

+

Zp

nL+1 nT+1

nL+2 nT+2

Zs/2

Zs/2

+ L

Vn

VnT

−

−

nL

Zs

Zp

nL+1

nT

nT+1

FIGURE 2.5 Periodic structure where the reference planes of the symmetric T-circuit unit cell (vertical dashed lines) and asymmetric L-circuit unit cell (vertical dotted lines) are indicated. The superscripts L or T in the port variables and reference planes are used to differentiate between the asymmetric and symmetric unit cells. Notice that the port currents do not vary by shifting the reference planes InL = InT , contrary to the port voltages VnL VnT . To make the reference planes of the symmetric unit cells accessible, it suffices to split the series impedance into two identical impedances equal to Zs/2.

B = Zs + C=

Zs2 4Zp

1 Zp

2 28b 2 28c

whereas the following parameters apply to the asymmetric unit cell: A=1+

Zs Zp

2 29a

B = Zs

2 29b

1 Zp

2 29c

C=

D=1

2 29d

59

THE TRANSFER MATRIX METHOD

Introducing the elements of (2.28) in (2.27), the Bloch impedance corresponding to the structure described by the symmetric unit cell is found to be Zs 1 + ZB± = ±

Zs 4Zp

Zs 1+ 2Zp

=±

2

Zs Zs + 2Zp 2 2

2 30

−1

and it is real in the propagation regions if Zs and Zp are purely reactive impedances. For the structure composed by a cascade of the asymmetric L-circuit unit cells, the Bloch impedance is derived from (2.26) and (2.29)

ZB± = −

2Zs Zs Zp

Zs 2+ Zp

=±

2

Zs Zs Zs + 2Zp + 2 2 2

2 31

−4

and it is a complex number in the region where wave propagation is allowed. As mentioned before, according to the usual definition of the positive current for forward and backward traveling waves, the two solutions of the Bloch impedance must be actually expressed as follows:

ZB± =

Zs Zs Zs + 2Zp ± 2 2 2

2 32

Expression (2.32) indicates that the real part of the Bloch impedance is identical to that of the symmetric structure (2.30), as expected. Notice that the imaginary part is the impedance that must be cascaded to the truncated periodic line (+Zs/2 and − Zs/2 in the load and source planes, respectively) in order to transform the finite asymmetric periodic structure to a symmetric network. Therefore, the complex Bloch impedance in the propagation regions of lossless asymmetric structures indicates that to match a load/source to the line it is necessary to series connect a reactive impedance, able to compensate the effects of asymmetry, plus the required resistive part, given by the absolute value of (2.30). In the circuit of Figure 2.5, it is remarkable that the dispersion relation, given by (2.22), is insensitive to the position of the reference planes of the unit cell. Either by using the A and D parameters given by (2.28) or (2.29), expression (2.22) is found to be cos βl = 1 +

Zs 2Zp

2 33

60

ARTIFICIAL TRANSMISSION LINES BASED ON PERIODIC STRUCTURES

2.3.3 Effects of Asymmetry in the Unit Cell through an Illustrative Example At this point, it is interesting to provide an example to illustrate the effects of asymmetry of the unit cell in the behavior of the structure. Let us consider the ladder network corresponding to the circuit model of a lossless conventional transmission line, where each unit cell describes a section of the transmission line that must be electrically short (Figure 2.6). For the symmetric unit cell, the Bloch impedance, inferred from (2.30), is

ZB =

L ω2 1− 2 C ωo

2 34

where ω is the angular frequency, and ωo = 2 LC is a cutoff frequency. Above this cutoff frequency, the Bloch impedance is purely imaginary, and propagation is not allowed. The network of Figure 2.6 is indeed a low-pass filter, whereas a lossless transmission line is an all-pass structure. The discrepancy is explained because the network of Figure 2.6, where L and C model the per-section inductance and capacitance of the transmission line, is valid at low frequencies (the wavelength must be much larger than the considered line section length). Nevertheless, for frequencies satisfying ω ωo, the Bloch impedance coincides with the well-known characteristic impedance of the line (expression 1.9). For the asymmetric unit cell (L-network), the Bloch impedance, given by expression (2.32), is ZB± =

L ω2 1 1 − 2 ± j Lω C 2 ωo

2 35

As frequency decreases, the imaginary part of the Bloch impedance also decreases. Therefore, at sufficiently low frequencies, where Lω/2